Gaussian-based computations in molecular science

Journal of Molecular Structure (Theochem) 671 (2004) 1–21 www.elsevier.com/locate/theochem

Review

Gaussian-based computations in molecular science A.F. Jalbouta,*, F. Nazarib, L. Turkerc a

Department of Chemistry, The University of New Orleans, New Orleans, LA 70148-2820, USA b Department of Physics, Dillard University, New Orleans, LA 70112, USA c Middle East Technical University, Department of Chemistry, 06531 Ankara, Turkey Received 4 February 2003; accepted 30 April 2003

Abstract A review article of over 300 hundred references describing the background and recent advances in the development and application of Gaussian based methods is presented. q 2004 Published by Elsevier B.V. Keywords: Review article; Gaussian-based methods; Development; Application

1. Background and recent advances Computational chemistry is focused on obtaining results relevant to chemical problems, not directly on developing new theoretical methods. There is of course a strong interplay between traditional theoretical chemistry and computational chemistry. Developing new theoretical models may enable new problems to be studied, and results from calculations may reveal limitations and suggest improvements in the underlying theory. Depending on the accuracy wanted, and the nature of the system at hand, one can today obtain useful information for systems containing up to several thousand particles. One of the main problems in computational chemistry is selecting a suitable level of theory for a given problem, and being able to evaluate the quality of the obtained results. A molecule may be considered as a number of electrons surrounding a set of positively charged nuclei. The Coulombic attraction between these two types of particle forms the basis for atoms and molecules. The potential between two particles with charges qi separated by a distance rij is (in suitable unites) given by qi qj ð1Þ Vij ¼Vðrij Þ¼ rij Besides the interaction potential, an equation is also needed for describing the dynamics of the system, i.e. how * Corresponding author. Fax: þ 1-504-816-6860. E-mail address: [email protected] (A.F. Jalbout). 0166-1280/$ - see front matter q 2004 Published by Elsevier B.V. doi:10.1016/S0166-1280(03)00347-6

the system evolves in time. In classical mechanics this is Newton’s second law (F is the force, a is the acceleration, r is the position vector and m the particle mass). F ¼ ma 2

ð2Þ

dV d2 r ¼m 2 dr dt

ð3Þ

Electrons are very light particles and cannot be described by classical mechanics. They display both wave and particle characteristics, and must be described in terms of a wave function, C: The quantum mechanical equation corresponding to Newton’s second law is the time-dependent Schro¨dinger equation (h is Planck’s constant divided by 2p). H C ¼ ih

›C ›t

ð4Þ

If the Hamilton operator, H; is independent of time, the time dependence of the wave function can be separated out as a simple phase factor. Hðr; tÞ ¼ HðrÞ

ð5Þ

Cðr; tÞ ¼ CðrÞe2iEt=h

ð6Þ

HðrÞCðrÞ ¼ ECðrÞ

ð7Þ

The time independent Schro¨dinger equation describes the particle-wave duality, the square of the wave function giving the probability of finding the particle at a given position. For a general N-particle system the Hamilton operator contains kinetic ðTÞ and potential ðVÞ energy for all

2

A.F. Jalbout et al. / Journal of Molecular Structure (Theochem) 671 (2004) 1–21

particles. H ¼T þV T¼

N X

ð8Þ

Ti ¼ 2

i¼1

72i

¼

V¼

N X h 2 2 7 2mi i i¼1

›2 ›2 ›2 þ 2 þ 2 2 ›x i ›y i ›z i

N X N X

ð9Þ !

Vij

ð10Þ

ð11Þ

i¼1 j.1

where the potential energy operator is the Coulomb potential (Eq. (1)). As nuclei are much heavier than electrons, their velocities are much smaller. The Schro¨dinger equation can therefore to a good approximation be separated into one part which describes the nuclear wave function, where the energy from the electronic wave function plays the role of a potential energy. This separation is called the Born-Openheimer (BO) approximation. The electronic wave function depends parametrically on the nuclear coordinates; it depends only on the position of the nuclei, not their momenta. The picture is that the nuclei, not their momenta. The picture is that the nuclei move on potentials energy surfaces (PES), which are solutions to the electronic Schro¨dinger equation. Denoting nuclear coordinates with R and subscript n; and electron coordinates with r and e; this can be expressed as follows. Htot Ctot ðR; rÞ ¼ Etot Ctot ðR; rÞ

ð12Þ

Htot ¼ He þ Tn

ð13Þ

He ¼ Te þ Vne þ Vee þ Vnn

ð14Þ

Ctot ðR; rÞ ¼ Cn ðRÞCe ðR; rÞ

ð15Þ

He Ce ðR; rÞ ¼ Ee ðRÞCE ðR; rÞ

ð16Þ

ðTn þ Ee ðRÞÞCn ðRÞ ¼ Etot Cn ðRÞ

ð17Þ

The BO approximation is usually very good. For the hydrogen molecule the error is of the order of 1024, and for systems with heavier nuclei, the approximation becomes better. It is only possible in a few cases to solve the electronic part of the Schro¨dinger equation to an accuracy of 1024, i.e. neglect of the nuclear – electron coupling is usually only a minor approximation compared with other errors. Once the electronic Schro¨dinger equation has been solved for a large number of nuclear geometries (and possibly also for several electronic states), the PES is known. This can then be used for solving the nuclear part of the Schro¨dinger equation. If there are N nuclei, there are 3N coordinates that define the geometry. Of these coordinates three describe the overall translation of the molecule, and three describe the overall rotation of the molecule with respect to three axes. For a linear molecule, only two coordinates are necessary for describing the rotation. This leaves 3n 2 6ð5Þ coordinates to describe the internal movement of the nuclei, the vibrations, often chosen to be

‘vibrational normal coordinates’. It should be noted that nuclei are heavy enough for quantum effects to be almost negligible they behave to a good approximation as classical particles. Indeed, if nuclei showed significant quantum aspects, the concept of molecular structure (i.e. different configurations and conformations) would not have any meaning; the nuclei would simply tunnel through barriers and end up in the global minimum. Furthermore, it would not be possible to speak of a molecular geometry, since the Hisenberg uncertainty principle would not permit a measure of nuclear positions to accuracy much smaller than the molecular dimension. Methods aimed at solving the electronic Schro¨dinger equation are broadly referred to as ‘electronic structure calculations’. An accurate determination of the electronic wave function is very demanding. Constructing a complete PES for molecules containing more than 3 –4 atoms is virtually impossible. Consider for example mapping the ˚ over say a 1 A ˚ range PES by calculating Ee for every 0.1 A (a very coarse mapping). With three atoms there are three internal coordinates, giving 103 points to be calculated. Already four atoms produces six internal coordinates, giving 106 points, which only can be done by a very determined effort. Larger systems are out of reach. Constructing global PESs for all but the smallest molecules is thus impossible. However, by restricting the calculations to the ‘chemically interesting’ part of the PES it is possible to obtain useful information. The interesting parts of a PES are usually nuclear arrangements which have low energies. For Example, nuclear movements near a minimum on the PES, which corresponds to a stable molecule, are molecular vibrations. Chemical reactions correspond to larger movements, and may in the simplest approximation be described by locating the lowest energy path leading from one minimum on the PES to another [1]. The electronic energy for a given nuclear configuration can be calculated by the following methods: (a) Self-consistent Field Theory (b) Force Field method (Molecular Dynamic) (c) Semi-empirical methods (ZDO, NDDO, INDO, CNDO, MINDO, MNDO, AM1, PM3, EHT) (d) Electron Correlation Methods (CI, MCSCF, MPn, CC, CCSD, G1, G2, G3, CBS-4, CBS-Lq, CBS-Q, CBS-QB3 and CBS-APNO) (e) Density Functional Methods

2. The terminology of modern Gaussian based methods Ab-initio methods try to derive information by solving the Schro¨dinger equation without fitting parameters to experimental data. Actually, ab-initio methods also make use of experimental data, but in a somewhat more subtle fashion. Many different approximate methods exist for solving the Schro¨dinger equation, and the one to use for


a specific problem is usually chosen by comparing the performance against known experimental data. Experimental data thus guides the selection of the computational model, rather than directly entering the computational procedure. One of the approximations inherent in essentially all ab-initio methods is the introduction of basis set. There are two types of basis functions (also called Atomic Orbitals, AO, although in general they are not solutions to an atomic Schro¨dinger equation) commonly used in electronic structure calculations: Slater type orbitals (STO) and Gaussian type orbitals (GTO). STO have the functional form [2]

xj;n;l;m ðr; u; wÞ ¼ NYl;m ðu; wÞr n21 e2jr

ð18Þ

N is normalization constant and Yl;m is the usual spherical harmonic functions. GTO [3] can be written as:

xj;n;l;m ðr; u; wÞ ¼ NYl;m ðu; wÞr 2n2221 e2jr

2

ð19Þ

The r 2 dependence in the exponential makes the GTO inferior to the STOs in two aspects. At the nucleus the GTO has zero slope, in contrast to the STO which has a ‘cusp’ (discontinuous derivative), and GTOs have problems representing the proper behavior near the nucleus. The other problem is that the GTO falls off too rapidly far from the nucleus compared with an STO, and the ‘tail’ of the wave function is consequently represented poorly. Both STOs and GTOs can be chosen to form a complete basis, but the above considerations indicate that more GTOs are necessary for achieving a certain accuracy compared with STOs. Combining the full set of basis functions, known as the primitive GTOs (P GTOs), in a smaller set of functions by forming fixed linear combinations is known as basis contraction, and the resulting functions are called contracted GTOs (C GTOs). 2.1. Classification of basis sets The complete set of basis functions is used to represent the molecular orbitals. In most cases, the basis set simply consists of the relevant exponents and coefficients of Gaussian functions. The following basis sets are the popular. STO-3G [4 –5], 3-21G [6 – 10], 6-21G [6 –7], 4-31G [11 – 14], 6-31G [11 – 15], 6-31G þ , 6-31Gþ þ , 6-31G (d0 ), 6-31G (d0 ,p0 ), 6-311G, D95V: Dunning/Huzinaga valence double-zeta [16], D95: Dunning/Huzinaga full double zeta SHC: D95V on first row, Goddard/Smedley ECP on second row [16 –17], Also known as SEC,CEP-4G: Stevens/Basch/ Krauss ECP minimal basis,CEP-31G: Stevens/Basch/Krauss ECP split valance, CEP-121G: Stevens/Basch/Krauss ECP triple-split basis [18 –20], LanL2MB: STO-3G [5,21] on first row, Los Alamos ECP plus MBS on Na –Bi [5,21,22], LanL2DZ: D95 on first row [16], Los Alamos ECP plus DZ on Na – Bi [22,23], SDD: D95V on the first row [16] and

3

Stuttgart/Dresden ECP’s on the remainder of the periodic [24 – 48], cc-pVDZ, cc-pVTZ, cc-pVQZ, cc-pV5Z, ccpV6Z: Dunning’s correlation consistent basis sets [49 – 53] (double, triple, quadruple, quintuple-zeta and sextuple-zeta, respectively). These basis sets have had redundant functions removed and have been rotated [52] in order to increase computational efficiency. 2.2. Methods HF. Hartree – Fock calculation [55 – 57]. Møller– Plesset theory. A way of introducing electron correlation into a calculation by perturbing the Hamiltonian and calculating the energy to different orders of expansion. MP2-Møller – Plesset theory truncated at second order. MP4, MP4 (SDQ) and MP4 (SDTQ)-Møller – Plesset theory truncated at 4th order. MP5 truncated at fifth order [58 – 71]. Configuration Interaction (CI). A way of introducing electron correlation into a calculation by mixing in an optimum amount of different electronic configurations. The ‘weightings’ of these electronic configurations are called the CI coefficients. CIS. A way of calculating on excited states using singleexcitation CI (CI-Singles) [72 –74]. CID. CI in which the extra electronic configurations are restricted to those that can be obtained from double excitations of the reference wave function. CISD. Like CID but both single and double excitations are allowed [75,76]. Multi-Configuration SCF (MCSCF). A multi-configuration treatment in which the CI coefficients are optimized for all possible electronic configurations within an active space (i.e. a small subset of the full range of molecular orbitals). Complete Active Space SCF (CASSCF). This is a multiconfiguration treatment in which both the molecular orbital coefficients and the CI coefficients are optimized within the active space (ie a small subset of the full range of molecular orbitals) [77 – 81]. Generalized valance bond (GVB). This method is for calculating a perfect-pairing General Valence Bond (GVBPP). QCISD. This method is a Quadratic CI calculation including single and double substitutions. This keyword requests only QCISD and does not include the triples correction. CCD, CCSD. (Coupled Cluster) coupled cluster calculations, using double substitutions from the Hartree – Fock determinant for CCD or both single and double substitutions for CCSD. CC is a synonym for CCD [82 –87]. QCISD (T) or (TQ). This is a Quadratic CI calculation including single and double substitutions with an energy contribution from triples for QCISD (T) and quadruples added for TQ [88 –90]. BD. This method is used for a Brueckner Doubles calculation [91,92].

4


OVGF. These methods request an Outer Valence Green’s Function (propagator) calculation of correlated electron affinities and ionization potentials. Either R or U must be specified with OVGF. The unrestricted code requires a two-electron integral file as well as kiakbcl transformed integrals [92 – 100]. Density Functional Theory (DFT). A way of introducing electron correlation by adapting the expression for the density of a uniform electron gas to the molecular orbital environment. B3lyp. A density functional method due to Lee, Yang and Parr which incorporates a three-parameter functional due to Axel Becke. BLYP. A density functional method developed by Becke, Lee, Yang and Parr. BP86-A density functional method developed by Becke and Perdew in 1986 [101 –107]. G1, G2, G2MP2, and G3. These methods request the Gaussian-1 (more colloquially known as G1) and Gaussian2 (G2) methods for computing very accurate energies. G2MP2 requests the modified version of G2 known as G2 (MP2), which uses MP2 instead of MP4 for the basis set extension corrections and is nearly as accurate as the full G2 method at substantially reduced computational cost. All of these methods are complex energy computations involving several pre-defined calculations on the specified molecular system. All of the distinct steps are performed automatically when one of these keywords is specified, and the final computed energy value is displayed in the output. No basis set keyword should be specified with these keywords. G3 is a refinement of previous energy models G1 and G2, which have been under development for more than a decade. These methods are complex energy computations involving several to many pre-defined calculations on the specified system. The main computational steps are shown in Figure 1. In addition to the standard type of basis sets, a large basis (G3large), which permits a flexible description of the whole space with inner shells, is added. This basis is so large that only MP2 computations are reasonably possible. Geometrical structure in the G3 model are determined at the MP2/6-31G (d) level. This is followed by a sequence of single-point calculations which aim to estimate the results of a potential QCI/G3 large energy, by assuming that effects of some of the improvement steps can be treated additively [108 – 112].

These methods refer to the CBS-4, CBS-q (i.e. Lq for ‘little q’), CBS-Q], CBS-Q//B3 and CBS-APNO methods, respectively. No basis set should be specified with any of these methods [113 – 118]. TD. This method requests an excited state energy calculation using the time-dependent Hartree – Fock or DFT method [119 – 120]. INDO. This method keyword requests an excited state energy calculation using the ZINDO-1method Note that ZINDO calculations must not specify a basis set [121 –128]. Semi-Empirical Methods (ground state). Modern semiempirical methods involve the generation of a selfconsistent field. In this, the electron distribution is made a function of the kinetic and potential energy of the electrons. In part, the potential energy depends on the distribution of the electrons; that is the energy of the electrons is a function of the inter-electron interactions. In the LCAO methods, these interactions are limited to two-electron interactions, as described by the Roothaan equations. Because evaluation of the two-electron interaction energies is very difficult, the development of SCF methods has evolved into two branches, one of which, the ab-initio methods, attempt the non-empirical evaluation of these terms. The other branch, semi-empirical methods, avoids even attempting the evaluation of the integrals involved; instead they are replaced with approximations [129]. CNDO [130], INDO [131], MINDO/3 [132,133], MNDO [132,134 – 141], AM1 [132,137,138,142,143] and PM3 [144,145].are currently available semi-empirical approaches. Molecular dynamic method. Molecular Dynamics (MD) is used to simulate the motions in many-body systems. By MD we mean the brute force solution of the classical equations of motion subject to pertinent boundary conditions. In a typical MD simulation one first starts with an initial state of the system. This is a list of the coordinates required to define the positions of all of the particles in the system and the momenta conjugate to these coordinates. Thus for an N particle system there will be 3N coordinates and 3N conjugate momenta and the mechanical state of the system will thus be specified by a 6N-dimensional vector G ¼ ðx1 ; x2 ; …; x3N ; P1 ; P2 ; …; P3N Þ: After sampling the initial state one must numerically solve Hamiltonian’s equations of motion:

Basis

HF

MP2

MP4

QCISD (T)

x_ i ¼

6-31G(d) 6-31 þ G (d) 6-31G(2df,p) G3large

Freq

Opt-1 4 6 8

2 5 7

3

Complete Basis Set Methods. (CBS-4, CBS-Lq, CBS-Q, CBS-QB3 and CBS-APNO). These methods specify the various Complete Basis Set (CBS) methods of Petersson and coworkers for computing very accurate energies.

›H › Pi

p_ i ¼ 2

›H ›xi

ð20Þ ð21Þ

subject to the initial conditions. Thus in all but a few simple systems, one must use finite difference approximations or integrators to solve the equations of motion. MD can be used to simulate the properties of bulk materials. MD calculations use the AMBER: The AMBER force field is described in Ref. [146]. The DREIDING force


field is described in Ref. [147] and UFF force field is discussed in Ref. [148]. also UFF uses charges which is calculated by the method of charge equilibration [149]

3. Geometrical optimization implementation in Gaussian and complete basis set computations One of the central tasks of computational chemistry is the reliable prediction of molecular structures. It is routine now to carry out geometry optimizations on systems consisting of up to, say, 50 atoms. These methods can be categorized as follows: 3.1. Wave function based The search for accurate electronic wave functions of polyatomic molecules uses mainly the MO method. A full theoretical treatment of a polyatomic molecule involves calculation of the electronic wave function for a range of each of parameters, the equilibrium bond distances and angles are then found as those values that minimize the electronic energy (including nuclear repulsion). The ab-initio methods are distinguished by the specific approach to solving the Hartree – Fock equations, the treatment of electron correlation, and above all, by the basis set used. The semi-empirical methods are characterized by the specific integrals encountered in solving the Hartree –Fock equations that are being approximated and by the manner of parametrization of those integrals. Various highly accurate ab-initio composite methods of Gaussian-n (G1,G2,G3), their variations (G2 (MP2), G3 (MP2), G3//B3LYP, G3 (MP2)//B3LYP), and complete basis set (CBS-Q, CBS-Q//B3LYP) series of models were applied to compute geometrical structure, thermodynamic properties of different molecules and reactions [150]. A new ab-initio molecular dynamics (AIMD) method is described [151] which is based on dynamical propagation of the electronic density matrix elements via extended Lagrangian formalism using Gaussian orbitals [152]. This AIMD methodology, which will be available in a forthcoming release of the Gaussian software package, promises to allow for more accurate AIMD studies of large, chemically reactive systems. By using these improvements the gas-phase thiocarbonyl transfer reactions X2 þ RCð2 SÞ ZRCð2 SÞX þ Z2 have been investigated with X, Z ¼ Cl, Br and R ¼ H, CH3 at the MP2 and G2(þ ) levels using the MP2/6-311 and G** optimized geometries [153]. And B3LYP method with 6-31 þ G (d,p) and cc-pVTZ basis sets is used for exploring the mechanism of s-trans2,4-hexadiene triplet interconversion [154]. Also the ab-initio direct dynamics method at the G2//UQCISD/6311 þ G (d,p) level is employed to study the hydrogen abstraction reaction [155]. The direct hydrogen abstraction process investigated at the UQCISD/6-311 þ G (d,p) and G2//UQCISD levels of

5

theory [156]. The reactions of CF2 (X1A1) radicals with O (3P) and H atoms have been studied by using by the G2-level ab- initio MO calculation [157]. The potential energy surfaces (PES) of reaction of S with C2H were computed by means of the G2, G2 (QCI), and CBS-Q methods in the case of local minimum. The MR-AQCC/aug-cc-pVTZ method was used as the basic level of computation; spin-orbit interactions and basis set superposition corrections were also taken into account [158]. The bond dissociation energies for silyl peroxides were calculated at the G2 and CBS-Q levels of theory [159]. Failure of a nominally high-accuracy ab-initio calculation of a modified G2 type for a reaction that is of considerable importance in hydrocarbon combustion is reported. The failure is traced to the reliance on single-reference methods in the treatment of electron correlation [160]. The possible products and intermediates of the title reaction were studied by means of the Gaussian-3 model calculations and MRCISD method [161]. The composite ab-initio methods CBS-Q and G3 (MP2) are used to analyze PES for vinyl radical addition reaction [162].Substituted Vinyl cations (H2C: C(þ )R), with R ¼ H, CH:CH2, CH3, F, and Cl, and their neutral precursors (H2C:CHR) have been studied using various quantum chemical methods to analyze the influence of these substituents on the thermodynamic stability and electronic properties. B3LYP data obtained with various basis sets are compared to those of post-HF computations, including MP2, MP4 (SDQ), and QCISD (T) computations and the CBS-Q model chemistry. The NBO and AIM population analysis methods are used for analysis of the electronic properties. The geometry, stability, and electronic structure of the vinyl cations under study are already accurately described at the B3LYP/6-311 þ G(d,p)//B3LYP/6-311G(d,p) and MP2/6311 þ G(d,p)//MP2/6-311G(d,p) levels of theory [163]. The gas-phase structures of the dilithiosalts of (N-methyl) dimethylsulfoximine were calculated by abinitio methods employing different levels of theory (HF, MP2, B3LYP and CBS). A comparison between DFT and CBS-Q calculations is made and reveals no significant discrepancies in the structures. The monolithium and Li-free dianions and their complexation energies to the Li cations were also calculated. NBO analyses were carried out to reveal donor –acceptor interactions in these dilithio salts [164]. 3.2. DFT based DFT based methods replace the complicated N-electron wave function with its dependence on 3N spatial plus N spin variables by a simpler quantity, such as the electron density. Systematic testing of modern density functional methodology for a broad variety of chemically motivated questions is of paramount interest, because the critical evaluations of such results provide the only means to assess the reliability of current DFT methods. This type of research

6


has led already to a substantial body of experience. Some selected application of this method is given in the following. Some of the methods involved in carrying out density functional calculations within the framework of localized basis sets, specifically those of GTO are reviewed by Briddon et al. Particular emphasis is placed on the methods used in the AIMPRO (Ab Initio Modeling Program) code [165]. A simple, rigorous and efficient combined real space/reciprocal space approach for electronic structure calculations within density functional based AIMD calculation is presented. Specifically, a novel combination of discrete variable presentations (DVRs) commonly used in reactive scattering and a plane wave basis is employed [166]. Computational studies on Alþ (H2O)n, n ¼ 6 – 9; and HAlOHþ(H2O)n – 1, n ¼ 6 – 14 by the DFT based AIMD method is reported. This method employs a planewave basis set with pseudopotentials, and also by conventional methods with Gaussian basis sets is reported. The mechanism for the intracluster H2 elimination reaction is explored [167]. Semi-empirical (AM1,PM3), ab-initio (RHF type), and density function theory (B3LYP type) MO methods are selected from Gaussian 98 package to study structure of the HNIW molecule [168].The equilibrium geometry and molecular structure of styrene has been optimized with HF/6-31G*,MP2/6-31G* and BLYP/6-31G* methods. At same time, using BLYP/6-31G* method, the harmonic frequency of styrene and its isotopemers, the bond energy of C-D bond (with ZPE correction), the intensity of IR spectrum are studied [169]. Geometry optimizations with Becke’s hybrid functional (B3LYP) and the 6-31 þ G(d,p) basis set gave good-quality equilibrium structures of several C2H6OS molecules and C2H7OSþ ions which did not improve significantly by B3LYP calculations using the larger 6-311 þ G (2df,2p). Also the proton affinities of these compounds are calculated. In general, proton affinities were overestimated by B3LYP calculations and underestimated by MP2 calculations with all basis sets used. Empirical averaging of the B3LYP and MP2 values, obtained from calculations with the 6-311 þ G(2df, 2p) basis set, provided an improved agreement with experimental. or accurate G2(MP2) proton affinities [170].Comparison of the calculated values in the thermodynamic cycle with experimental shows that the B3PW91 results are within the error bars or slightly outside them for all the reactions except the thermal decomposition of CF3OOCF3 [171]. The activation barriers for computationally difficult radical abstraction reactions with small radicals were studied with Becke’s 88 DFT and G1 and G2 ab-initio methods. Although many DFT methods produced negative activation barriers, Becke’s 88 DFT methods generated activation barriers, which are even closer to experimental. Therefore, it was suggested that HFB/6-311G (2d, 2p) should be the method of choice

for the DFT study of hydrogen radical abstraction reactions with small radicals [172]. A good description of the electronic structure, consistent with the experimental data, was achieved using DFT simulations for (OEC)Co(C6H5), where OEC is the trianion of 2,3,7,8,12,13,17,18-octaethylcorrole. Both the experimental. and calculated data support the conclusion that there is significant spin density on both the macrocycle and in the cobalt dyz orbital. [173]. The energetics of the gas-phase SN2 reactions,Y2 þ CH3X ! CH3Y þ X2 (where X, Y ¼ F, Cl, Br), were studied using (variants on) the recent W1 and W2 ab-initio computational thermochemical methods. These calculations involve CCSD and CCSD(T) coupled cluster methods, basis sets of up to spdfgh quality, extrapolations to the one-particle basis set limit, and contributions of inner-shell correlation, scalar relativistic effects, and (where relevant) first-order spin-orbit coupling Only the ‘half-and-half’ functionals BH and HLYP and mPWH and HPW91, and the empirical mPW1K functional, consistently find all required stationary points; the other functionals fail to find a transition state in the F/Br case. BH and HLYP and mPWH and HPW91 somewhat overcorrect for the tendency of B3LYP (and, to a somewhat lesser extent, mPW1PW91) to underestimate barrier heights. The Becke97 and Becke97-1 functionals perform similarly to B3LYP for the problem under study, while the HCTH and HCTH120 functionals both appear to underestimate central barriers [174]. The PES for the reaction of atomic H with propyne was studied at the G3//UB3LYP/6-31G (d) level of theory [175]. The PES of COþ 2 is investigated with HF, MP2, MP4, CBS-Q, G1, G2MP2, G2, G3B3, and B3LYP/6-311þ þ G (3df, 3pd) methods. DFT shows the lowest dissociation channel of this compound to be the formation of COþ þ Oþ and to have a barrier of around 2 eV as well as dissociation energy of around 2 3.2 eV. It is proposed that with enough correlation, it is possible to accurately predict the energies of dissociation [176]. DFT is used for modifying the complete basis set model chemistries CBS-4 and CBS-Q for the geometry optimization step of these methods. The accuracy of predicted bond dissociation energies and transition state barrier heights was investigated based on geometry optimizations using the B3LYP functional with basis set sizes ranging from 3-21G(d,p) to 6-311G(d,p) [177]. Using three hybrid DFT methods (B3LYP, B3P86, B3PW91) and the G1, G2, G2MP, and CBS-Q ab-initio computational approaches the hydrogen radical association reaction with carbon monoxide was studied [178]. Finally, the ability of DFT and the Gaussian-3 method to match benchmark results for adiabatic excitation energies of C2H2 is investigated [179].


4. Molecular stability The quantum chemical calculations now are indispensable tools for analyzing the experimental data and for planning new experiments. The theoretical stability sequences can be obtained by full geometry optimizations, and additionally, by an isodesmic reaction approach. In the latter case the total interaction energy between the various substituents is determined from the sum of single interaction energies between each two adjacent substituents, which are calculated from corresponding interaction-forming reactions. It is shown that the isodesmic reaction approach provides a quick and easy, but nevertheless reliable means for estimating relative stabilization energies from a simple increment system. It also provides a valuable tool for a discussion of the importance of the single contributions. Expectedly, the number and kind of hydrogen bonds is clearly the first and most decisive factor that governs the conformational stabilities. Application of different methods for investigating the molecular stability are given in the following. Employing both MP2 and DFT (B3LYP) methods the triplet PES for the N(4S) þ CH2F(2A0 ) reaction has been studied. The energies of the involved species have been refined using the G2, CBS, and CCSD (T) methods [180]. G2-type ab-initio calculations on model ions HO –[Mþ]– OH (M ¼ Li, Na, K, Cs), as well as arguments based on a simple Columbic interaction model, show that the bidentate stabilization energy drops rapidly as the size of the alkali cation increases [181]. Attempts to locate regions on the triplet approach surface where the singlet crosses to become the lower energy spin state were complicated by the difficulty of optimizing geometries within the composite G2 model. Preliminary efforts, however, indicate that such crossings occur at geometries higher in energy than separated:CH2 þ CO2, suggesting that their role should be relatively unimportant in this reaction [182]. G2 ab-initio and DFT calculations is applied to get a general picture of the acidity trends within Group 14 based and the gas-phase acidity of 3-methylcyclopropene at the allylic position according to their stabilities. The acid strength increases down the group, although the acidity differences between Ge and Sn derivatives are already rather small. Also acidities and homolytic O-H bond dissociation energies of CHnF3 – nOH and CHnF32nSH ðn ¼ 0 – 3Þ were calculated at the G2 level of theory [183 – 185]. The theoretical activation energies obtained using G2 (MP2), G2 (MP2,SVP)/B3-LYP and G2/CASSCF10-in-10 of Duan and Page [186]. The CBS-Q//B3LYP/6-31G (d,p) and G3(MP2) composite methods are utilized to calculating energies and the well depth and transition state calculated results is used for investigating the reaction path [187]. High-level ab-initio calculations at the G3 (MP2)//B3-LYP level have been used to study

7

carbomethoxychlorocarbene and related halogenocarbenes and carbonyl carbenes. Initial calculations at the more accurate W10 level on the subset CH2, HCCl, HCF, CCl2, and CF2 provide support for the reliability of G3 (MP2)//B3LYP for this type of problem [188]. The energies of six stationary points on the OH þ CO, HOCO, H þ CO2 PES were calculated using the G3 and CBS-QB3 methods [189]. Ab-initio calculations at the CBS-Q level of theory was done for the proton affinity of peroxyacetyl nitrate (PAN). This calculation revealed the complicated protonation chemistry associated with the unusual multifunctional structure of this molecule. Also optimized molecular structures, relative energies, decomposition energies, and proton affinities were determined for four chemically distinguishable PANHþ protomers. [190]. The probable reaction mechanism for the reaction of Me and chlorine monoxide radicals has been studied using the G2 MP2 method [191].

5. Transition state chemistry Transition state theory assumes that a reaction proceeds from one energy minimum to another via an intermediate maximum [192]. The transition state is the configuration which divides the reactant and product parts of the surface, while the geometrical configuration of the energy maximum is called transition structure. The reaction proceeds via a reaction coordinate. The reaction coordinate leads from the reactant to the product along a path where the energy is as low as possible, and the TS is the point where the energy has a maximum. Using the calculated property of transition state the rate constant can be calculated. Selected transition state calculations are given in the following part. Transition structures and energy barriers of the concerted prototypical cycloaddition reaction of 1,3-heterocumulenes (S:C:S, S:C:NR, RN:C:NR, and hetero analogs) to acetylene resulting in nucleophilic carbenes were calculated by G2(MP2) and CBS-Q ab-initio quantum chemistry and by DFT methods [193]. Also Single-point QCISD (T) and B3LYP calculations with large basis sets were performed to verify barrier heights on important transition states of direct reaction of acetylene with oxygen [194]. Hydrogen abstractions by O (3P) from a set of fluorinated methanes were studied using ab initio methods. Geometries of the reactants and transition states were optimized at the UMP2/ 6-311G** level of theory, and activation energies were calculated using a modified Gaussian 2 (G.cxa.2) theory [195]. Hydrocarbon radical reactions with oxygen are important in ignition and low temperature oxidation. Thermodynamic properties of reactants, products and transition states are determined by ab-initio methods at the CBS-Q and G2 levels of theory based on the optimized geometry using B3LYP/6-31G (d,p) DFT and isodesmic reaction analysis. Rate coefficients for reactions of

8


the energized adducts are obtained from canonical transition state theory [196]. The unimolecular dissociation of azomethane is studied by first sampling the reaction channels with a DFT based AIMD method and then by mapping out the reaction barrier and transition structure for each channel with Gaussian based ab-initio method at G3 and B3LYP/6-31G (d) levels [197]. The Arrhenius parameters for the high pressure limiting rate constant for the C – C bond scission of t-butoxy radicals have been computed from the properties of a transition state based on the results of G2 (MP2) ab-initio calculation [198]. By Using ab-initio MP2/6-311G** level of calculation the mechanism of addition and hydrogen abstraction reactions between NH2 (, X2B1) and C2H4 has been investigated. The transition states of the two reaction paths are obtained and verified by vibration analysis and IRC calculations [199]. Thermochemical properties of reactants, alkyl hydroperoxides (ROOH), hydroperoxy alkyl radicals, and transition states (TSs) are determined by ab-initio and density functional calculations. Enthalpies of formation of product radicals are determined using isodesmic reactions with group balance at MP4(full)/6-31G(d,p)//MP2(full)/631G(d), MP2(full)/6-31G(d), complete basis set model chemistries, CBS-q with MP2 (full)/6-31g(d) and B3LYP/6-31g(d) optimized geometries, and density functional (B3LYP/6-31g(d) and B3LYP/6-311 þ G(3df,2p)//B3LYP/6-31G(d)) calculations [200]. The HNO isomerization into HON was examined with a variety of DFT methods, including the B1LYP, B3LYP, and the MPW1PW91 method coupled to a relatively large 6311G(2d,2p) basis set. In addition, the CBS-Q, and G2 methods were used to further evaluate the barrier to isomerization, which is believed to occur through a threecenter cyclic transition state [201]. The Gaussian G3X and G3X2 methods was applied for investigating the thermochemistry, including the heats of formation of each of the adducts and the appropriate transition states of following reactions HOPO þ OH ! ( HO)2PO ! H2O þ PO2, HOPO þ H ! P(OH)2 ! H2O þ PO, (HO) 2PO þ H ! P(OH) 3 ! H 2O þ HOPO, and HOPO2 þ H ! (HO)2PO ! H2O þ PO2 [202]. Three procedures, density functional B3PW91/6311 þ G (2df, 2pd) and the ab- initio CBS-Q and CBSQB, were used: for computating, the transition states, activation barriers and overall enthalpy changes of the boron reactions [203]. Pople style theoretical model chemistries such as G3 or CBS-Q incorporates geometries and frequencies from Kohn – Sham density functional methods such as B3LYP, the modified model gives improved transition states for chemical reactions. A study of the convergence of calculated harmonic frequencies with basis set and level of correlation treatment shows that B3LYP/6-311G(d,p) and QCISD(T)/6-311 þ G(2df,p) calculations are particularly effective, reproducing the experimental frequencies for 10 stable molecules and radicals to within þ 44 and þ 15 cm,

respectively. The commonly used UHF/6-31G (d) and UMP2/6-311G (d,p) calculations have root-mean-square errors of 94 and 192.When applied to transition states for six chemical reactions, the UHF and UMP2 frequencies have RMS deviations from the QCISD(T)/6-311 þ G(2df,2pd) frequencies of 178 and 96 cm, respectively, whereas the B3LYP frequencies differ from the QCISD(T) values by only þ 24 cm. The transition state model chemistry employs the maximum G3 or CBS-Q energy along the B3LYP reaction path [204].

6. Cluster stability A major thrust in the development of high-tech materials can be described as atomic-scale engineering. In this process, materials are assembled on an almost atom by atom basis in order to obtain useful properties not found in naturally occurring substances. Applications of this new technology can be found in many industries, but nowhere are they more vigorously sought than in the electronics industry, where making devices like transistors smaller and more efficient quickly translates into new and improved consumer products—and significant new profits for manufacturers. There is currently a strong interest in the prospect of producing new materials consisting of small atomic clusters. Such clusterassembled materials may vary significantly from their crystalline counterparts. Mechanical, electronic, optical and other properties are expected to be different for such assemblies which should make them good potential candidates as new building materials for electronic devices. Also, the quantum effects which occur in such materials of finite size and dimension, lead to their special properties. Due to importance of these compounds great amount of researches devoted to this subject. Three of recent researches on this subject are given in the following. The ground state geometries and energetics of neutral and singly positive charged clusters are obtained using ultra-soft pseudo-potential plane wave method with generalized gradient approximations [205]. The stability properties of carbon clusters were studied using the advanced CBS-Q, G2, G3, G3B3 high level ab-initio methods as well as the DFT B3LYP/6-311þ þ G (3df, 3pd) method [206]. The ability of newly implemented methods such as CBS-QB3 and G3B3 to calculation of atomization energies of small clusters has been investigated. Although, these methods are limited in their convergence capabilities for larger polyatomic molecules for small carbon clusters (Cn, where n , 6) reasonable results can be obtained [207].


7. Molecular clusters 7.1. Van der Waals and H-bonded systems 7.1.1. Van der Waals interactions Dispersion forces, also referred to as London forces, are long-range attractive forces which act between separated molecules even in the absence of charges or permanent electric moments. These forces, which are purely quantum mechanical in nature, arise from an interplay between electrons belonging to the densities of two otherwise non-interacting molecule or atoms. Owing to their like electric charges, the molecular electron densities of two different systems repel each other if they come too close together. But at intermediate distances the motion of electrons in one unit induces slight perturbations in the otherwise evenly motion leads to a temporary dipole moment. The induced dipole moment, in turn, induces systems. In the asymptotic limit, this induced dipole-induced dipole attraction decays with the inverse sixth power of the intermolecular distance. The actual presence of interactions from higher order electric moments leads also to other terms like induced quadrupole –dipole, quadrupole – quadrupole interactions, etc. This effect is entirely due to electron correlation and the Hartree –Fock model is therefore not applicable to such situations. So the correlated methods must be taken in to consideration for studying these interactions [208]. Mayer calculated the binding energies of proton-bound clusters containing a nitrile at the HF, MP2, and B3-LYP levels of theory with a variety of basis sets. The results were compared with the previous work employing G2 based methods. Reliable binding energies could be obtained using the MP2/6-31 þ G (d) and B3-LYP/6-31 þ G(d) levels of theory [209]. The singlet and triplet structures of CHHeþ and doublet structure of CHHe2þ were studied with hybrid, local and gradient-correction DFT methods, as well as HF, MP2, G1, G2 and CBS-Q ab-initio methods. In all calculations, a 6-311þ þ G (3df, 3pd) basis set was used. High-level ab-initio calculations predict that singlet CHHeþ was approximately 26 kcal mol21 more stable than the corresponding triplet, while hybrid and gradient-correction DFT methods estimated that the energy preference is approximately 16 kcal mol 21 [210]. Ab-initio calculations [MP2,MP4SDTQ, and QCISD(T)] using different basis sets [6-31G (d,p), cc-pVXZ (X ¼ D, T, Q), and aug-cc-pVDZ] and DFT [B3LYP/6-31G(d,p)] calculations were carried out to study the OCS.(CO2)2 van der Waals trimer. The DFT has proved inappropriate to the study of this type of systems where the dispersion forces are expected to play a relevant role [211]. Alessandro et al. studied a complete cluster/ embedded cluster by means of ab-initio methods in the ONIOM scheme, as implemented in GAUSSIAN 98 code, of the reactivity towards water and ammonia of Ti(IV)

9

centers in zeolitic frameworks. All reported values being BSSE correction [212]. 7.1.2. Charge transfer interactions (Lewis complexes) A molecular complex between two molecules is an association somewhat stronger than ordinary van der Waals associations, of definite stoichiometry. The partners are very often already closed-shell (saturated structure) electronic structures. In loose complexes the identities of the original molecules are to a large extent preserved. The tendency to form complexes occurs when one partner is an electron acceptor (Lewis acid) and the other is an electron donor (Lewis base). Due to development of computational methods and computer facilities theoretical studies devoted to charge transfer complexes. For instance, the complexes between Niþ and a series of small nitrogen and oxygencontaining bases, were investigated by means of high-level G2 (MP2) ab-initio method and B3LYP density functional approach.The behavior of the bases investigated with respect to Niþ resembles closely the one they exhibit when the reference acid is Cuþ or Hþ. This can be taken as an indication of the non-negligible covalent character of the base – Niþ interactions [213]. Solling et.al. have employed the ab-initio calculations at the G2 level to investigate the ligand-exchange reactions between mono-adducts of the phosphenium ion (e.g. [H3N – PH2]þ) and simple first- or second-row Lewis bases (e.g. NH3). Virtually all the reactions proceeded without an intermediate barrier via a bis-adduct of the phosphenium ion with two Lewis bases (e.g. [H3N – PH2 –NH3]þ) was found [214]. Bagno et.al. studied the protonation site, aromaticity, charge distribution, and NMR properties of 3-aminothiophene, 3,4-diaminothiophene, aniline, and 1,2-benzenediamine by quantum chemical calculations both for the isolated and solvated species (in H2O and DMSO). For the isolated species (G3(MP2) level), the C-protonated form of aminothiophenes is more stable than the N-protonated form (by 5 –9 kcal mol21), whereas the stability order of the protonated forms of anilines is reversed, with a closer energy balance (2 –5 kcal mol21) [215]. Weakly bounded complex, DMS –OH, which from in the reaction of DMS with OH radical was studied by DFT-B3LYP Calculation. The results show that it is stabilized by 29.47 kJ mol21 compared to DMS þ OH from G3 calculation [216]. The basis for unprecedented non-covalent bonding between anions and the aryl centroid of electron-deficient aromatic rings was demonstrated by an ab-initio study of the interaction between 1, 3, 5-triazine and the fluoride, chloride, and azide ion at the MP2 level of theory [217]. Hybrid density functional theory (HDFT) computational studies of the complexation between aluminium and carbon monoxide was performed in order to determine its stability for experimental detection and accuracy of DFT computed values with some highly reliable ab-initio methods.

10


Four highly accurate ab-initio correlation calculations (G1, G2, G2 (MP2), and CBS-Q) agree that AlCO is more stable than its isomer AlOC, with Al-CO bond dissociation [218]. The G3 ab-initio and DFT calculations have been used for studying the reaction mechanisms between MH (M ¼ B, Al) and the H2S molecules. According to these calculations only one stable addition complex (HM: SH2, M ¼ B, Al) can be formed [219]. 7.1.3. H-bonded interactions (Bronsted complexes) An important category among weakly bound systems with binding energies smaller by one or two orders of magnitude is hydrogen bond or hydrogen Bridge. This type of bond is characterized by complex formation energy larger than just dipolar and dispersion interaction energies and by a H· · ·B bond length that is shorter than the sum of the van der Waals radii of H and B. B is usually atom more electronegative than hydrogen like O, F, N, or Cl. Although rather weak in nature, hydrogen bonds often have a decisive influence on the chemical properties of substances. In one of the earliest of all comparative DFT studies on hydrogen bonded systems, Latajka and Bouteiller, 1994, studied the hydrogen fluoride dimmer and applied a number of different functionals in combination with a variety of polarized Pople-type basis sets [220]. For the hydrogen bonded complexes (FH)2, (HCl)2, (H2O)2, FH· · ·CO, FH· · ·OC, FH· · ·NH3, ClH· · ·NH3, OH2· · ·NH3 , and H 3O þ · · ·H 2O, with binding energies ranging from 1.7 kcal mol 21 FH· · ·CO to 32.9 kcal mol21(H3Oþ· · ·H2O), Tuma, Boese, and Handy, 1999, compared results of several density functional methods with high level conventional wave function based data [221]. Using Gaussian ab-initio calculations involving DFT methods Roger reported the results on the stability and conformation for the 1:1 water – diol complex formed by ethane-1,2-diol, propane-1,2-diol, and L - and meso-butane2,3-diol. The relative stability of the intramolecular (internal) hydrogen bond in a range of diols ðn ¼ 2 – 6Þ; ethane-1,2-diol or other vicinal diols, which do not satisfy Popelier’s topological and electron density criteria based on the AIM theory of Bader. Based on these criteria it is unlikely that vicinal diols are in fact capable of forming an intramolecular hydrogen bond, in spite of geometric and spectroscopic data in the literature suggesting otherwise [222]. High-level G2 (MP2) ab-initio and B3LYP/6311 þ G (3df, 2p) density functional calculations have been carried out by Pablo et al. for intramolecular hydrogen bonds in a series of chalcogenovinylaldehydes. An analysis of these chalcogen –chalcogen interactions indicates that both, the electrostatic and the dative contributions are smaller for Se- than for Te-derivatives. In the latter, the electrostatic component clearly dominates when X ¼ O, while the opposite is found for sulfur-containing derivatives [223]. Hydrogen bonding between water and a series of small organic molecules was examined via electronic

structure calculations. Several computational methods were examined, including both a hybrid density functional procedure (Becke3LYP) and second-order Moller –Plesset theory (MP2) coupled with a double-j basis set augmented by diffuse polarization functions on heteroatoms. The agreement between Becke3LYP and MP2 energies was generally good, as was the agreement with energies obtained using more sophisticated and costly methods [224]. 7.2. Ringed systems The G2 (MP2) and B3LYP/6-31G (d,p) levels of theory are used for investigating of ring opening reactions of 2-furylcarbene (5a), 2-pyrrolylcarbene (5b), 1-cyclopenta1,3-dienylcarbene [225].For the first time various pure and hybrid density functionals with three larger basis sets have been used to study the troublesome and the most forensic molecule, SiC2. It is concluded that BLYP and B3LYP functionals have failed to predict the cyclic form of SiC2 as the most stable structure over its linear counterpart even with the most flexible basis set, 6-311 þ G(3df). On the contrary, the other two pure density functionals of Perdew, Perdew and Wang with larger basis sets correctly identify the minimum energy structure. Finally, the two hybrid functionals B3P86 and B3PW91 predict the cyclic isomer of SiC2 to be more stable than the linear and the increase in the size of the basis set increases the stability of the former [226]. Properties and ring opening reactions were studied for azaphosphirane and its P-Ph and W(CO)5 complex using DFT (B3LYP). Azaphosphirane has a relatively small N-inversion barrier of 10.8 kcal mol 21 and a high 56.8 kcal mol21 ‘turnstile’ P-inversion barrier. Its strain energy is 26.5 kcal mol21 at G3 (MP2) [227]. The strain energies for a small set of strained C3 – C6 cyclic hydrocarbons were calculated using homodesmotic reactions and ab-initio methods. The values calculated using the correlated methods B3LYP, MP2, QCISD (T), G1, G2, CBS-4, and CBS-Q were compared. Values determined at the QCISD (T)/6-311 þ G(2df,p) level were used as the ref. set, and values from the other methods were compared using the mean abs. deviation (MAD) as a measure of the performance for each one [228].

8. Accuracy in the prediction of chemical and physical properties 8.1. Vibrational frequencies Vibrational spectroscopy is of utmost importance in many areas of chemical research and the application of electronic structure methods for the calculation of harmonic frequencies has been of great value for the interpretation of complex experimental spectra. Numerous unusual molecules have been identified by comparison of computed


and observed frequencies. Another standard use of harmonic frequencies in first principles computations is the derivation of thermochemical and kinetic data by statistical thermodynamics for which the frequencies are an important ingredient. The theoretical evaluation of analytic second derivatives of the total energy with respect to cartesian coordinates [229,230]. At HF level, the calculated frequencies are commonly overestimated quite systematically by ca. 10%, which can be traced back to the missing electron correlation; basis set deficiencies and the neglect of anharmonicity. A great number of researches are devoted to frequency calculation; the following examples have been selected among them. The B3LYP method with a variety of basis sets as well as the very advanced CBS-Q, CBS-QB3, G1, G2MP2, G2, G3, and G3B3 ab-initio methods were used in order to analyze the vibrational spectra as well as ionization potentials of BeCH3, MgCH3 and CaCH3. Vibrational analysis by the B3LYP/6-311þ þ G (3df, 3pd) method of MgCH3 yields frequencies at 1084 and 502 cm21 with the experimental values being 1072 and 509 cm 21 , respectively. Comparisons to rotational constants were also made, with the B3LYP/6-311þ þ G (3df, 3pd) method, an example of this being the constant of CaCH3 calculated to about 7555 MHz compared to the experimental value of 7556 MHz [231]. Ab-initio MO methods at the CBS-Q level of theory have been used to study the structure, gas-phase acidities and vibrational spectra of formohydroxamic acid HC (:O) NHOH, and its sila derivatives HSi(:O)NHOH. The geometries of various tautomers and rotamers of formohydroxamic and silaformohydroxamic acids, their anions and protonated forms were optimized at the MP2 (FC)/6-31G (dag) level of theory [232]. The geometrical parameters, force field parameters, and vibrational frequencies of the enol forms of beta-diketones RCOCH2COR (R ¼ H, CH3, CF3) were calculated using the ab-initio MO LCAO SCF method involving broad bases of Cartesian Gaussian functions [233]. The structural and vibrational properties of the transition state of the N2O þ X (X ¼ Cl, Br) reactions have been characterized by ab-initio methods using DFT. Lesar et al. have employed Becke’s hybrid functional (B3LYP), and transition state optimizations were performed with 6-31G (d), 6-311G (2d, 2p), 6-311 þ G (3d, 2p), and 6-311 þ G (3df, 2p) basis sets. For the chlorine atom reaction the coupled-cluster method (CCSD(T)) with 6-31G(d) basis set was also used in addition, several highly accurate ab-initio composite methods of Gaussian-n (G1, G2, G3), their variations (G2(MP2), G3//B3LYP), and complete basis set (CBS-Q, CBS-Q//B3LYP) series of models were applied to compute reaction energetics [234]. Accurate torsional potentials and torsional barriers are derived for the 1,3-butadiene molecule using state-of-the-art coupled-cluster (CC) methods. The basis set effect, and the performance of different ab-initio methodologies

11

with respect to the CC calculations, were carefully addressed [235]. The geometries and vibrational frequencies of the HC3Cl and HC3Clþ isomers were obtained at the MP2 and B3LYP levels, whereas relative energies were computed at the MP4, QCISD (T), G2 and CCSD (T) levels. All levels of theory predict that the global minimum of HC3Cl is singlet chlorocyclopropenylidene, although singlet chlorovinylidenecarbene and singlet and triplet chloropropargylene are relatively close in energy [236]. Quantum chemical calculations were performed to derive structures and vibrational frequencies of the C –C bond dissociation of three fluorinated ethoxy radicals CF3CF2O, CF3CFHO and CF3CH2O has been investigated using the B3LYP/cc-pVTZ (þ 1) level. Higher level abinitio data such as critical energy barriers are provided by application of a modified G3 (MP2)-theory [237]. The amount of attention dedicated to the theoretical and experimental investigation of small cationic organometallic systems in the literature is very limited. In this research the B3LYP method with a variety of basis sets as well as the very advanced CBS-Q, CBS-QB3, G1, G2MP2, G2, G3, and G3B3 ab-initio methods are used to analyze the vibrational spectra as well as ionization potentials of BeCH3, MgCH3 and CaCH3. The need for further addition of experimental data to the archives for these systems is discussed, as well as recommendations for which theoretical methods are optimum for a particular result [238]. The C –C bond scission of three fluorinated ethoxy radicals CF3CF2O, CF3CFHO and CF3CH2O has been investigated using current theoretical methods. Quantum chemical calculations were performed to derive structures and vibrational frequencies at the B3LYP/cc-pVTZ (þ 1) level. Higher level ab-initio data such as critical energy barriers are provided by application of a modified G3 (MP2)-theory [239]. The reaction of atomic hydrogen with acetonitrile has been studied using the B3LYP and Gaussian-3 (G3) methods. The geometries and vibrational frequencies of various stationary points on the PES were calculated at the B3LYP level with the 6-311G (d,p) and 6-311þ þ G(2d,2p) basis sets [240]. In order to investigate the reaction of ClO with CH3OO, geometries are optimized and frequencies are estimated using MP2/6-31G(d,p), B3LYP/6-31G(d,p), or MP2/631G(d) level calculations and single point calculations for estimating of energy are performed with B3LYP/6311 þ G(3df,2p) and QCISDT/6-31G(d,p) and with composite methods of CBS-Q, G3/MP2 and density functional calculations [241]. Structures, vibrational frequencies and barrier heights characterizing the unimolecular reactions of primary and secondary alkoxy radicals up to pentoxy were determined by quantum chemical calculations. Several additive model chemistries such as CBS-4, CBS-q, CBS-Q and G2 (MP2, SVP) and B3LYP hybrid-functional were applied to obtain

12


information on the reactive PESs and to test their applicability to transition state structures [242]. Accurate and predictive values of the bond lengths and angles, dipole moments, and vibrational frequencies of BClþ 2 , BCl2 and BCl2 and the corresponding Al analogs are calculated by Becke’s three-parameter DFT (B3LYP) method with the augmented correlation consistent tripleand quadruple j (aug-cc-pVTZ and aug-cc-pVQZ, respectively) basis sets. The coupled-cluster singles doubles, and non-iterative triples (CCSD (T)) method employing the aug-cc-pVTZ basis set is also used to augment the B3LYP results. The ionization energies and electron affinities are also calculated by the B3LYP/aug-ccpV5Z method at the B3LYP/aug-cc-pVQZ geometry and the CCSD (T)/aug-cc-PVQZ method at the CCSD (T)/augcc-pVTZ geometry, as well as by the G1, G2, CBS-4, and CBS-Q methods [243]. 8.2. Ionization potentials Ionization potentials, the energy required to remove an electron from a bound state to infinite separation. Ionization processes of molecules have attracted much attention from the early 1970s onwards when photoelectron spectroscopy emerged as a new exciting experimental technique. This technique has severed as a testing ground for theoretical methods and nowadays, a strong synergy between experiment and theory renders this field of research highly efficient. Correlation effects are of particular importance for a proper description of ionization processes since the number of electrons change during the ionization. The following selected calculated ionization potentials show the ability of theoretical methods. B3LYP method is used for calculating ionization potential of PuH and Potential energy function and force constant and spectroscopic data for ground state of PuH, PuHþ and PuH2þ [244]. Electron ionization of C 3H6Oþ and C2H4Oþ are calculated with high level G2 ab initio calculations. Also threshold photoionization mass spectrometry is used as experimental device for deriving electron ionization. The results of two methods are both in excellent agreement [245,246]. High level G2(MP2) computational study provide purely computational thermodynamic data for the following processes: (i) Ionization of 1- and 2-Adamantyl and 1and 2-Adamantyl radicals (ii) Exchange of hydrogen atoms or hydride anions between Adamantyl radicals or cations and alkyl radicals or cations, respectively [247]. The various methods are compared using the G2/97 test set for calculating various ionization potentials, enthalpies of formation, electron affinities, and proton affinities. Methods include: empirical methods such as Benson group additivity and molecular mechanics; semi-empirical MO methods such as MINDO/3, MNDO, AM1, and PM3; DFT at B3LYP and BYLYP levels; and ab-initio

MO methods such as BAC-MP4, CBS-Q, G2, G3, and CCSD (T) [248]. The changes in electron populations that occur on ionization and the origin of the acidity of carboxylic acids and enols has been examined via ab-initio MO calculations at the MP2/6-31 þ G** and Becke3LYP/6-31 þ G** levels as well as with the CBS-4 (Complete Basis Set) and CBS-Q model chemistries [249]. 8.3. Electron affinities The electron affinity of a neutral system is the energy gained upon attaching an additional electron, thereby generating the corresponding anion. The addition of an electron does not in all cases leads to energetically more favorable anions. Rather, there are many atoms or molecules, where the energy of the anion is higher (i.e. less favorable) than that of the parent neutral, i.e. where the excess electron is not bound but will autodetach immediately. The species do not have positive, but a negative electron affinity. While there are sophisticated experimental techniques to probe the transient anionic species resulting from neutrals with negative electron affinities we will in the following only consider stable anions characterized by positive electron affinities. The computational prediction of electron affinities has always been a particularly difficult task for wave function based methods. Following selected topics show the ability of the theoretical methods for predicting the electron affinity. In the review by Rienstra– Kiracofe et al. the ability of computational chemistry methods to predict electron affinities is examined. A major portion of this work is a review (through January 2000) of 1101 experimental determined electron affinities via photoelectron techniques. Also included are reports of theoretical values for electron affinities of 163 atoms or molecules, as predicted by DFT from results by the Schaefer group. Nonetheless, discussion is not limited to DFT, and comparisons to the coupledcluster (CC), Gaussian-X (G2, G3), complete basis set (CBS-M, CBS-Q), and Weizmann-X (W1, W2) theoretical methods are made. Additional, DFT EA predictions for 53 atoms or molecules with unmeasured or uncertain experimental EAs are presented. The review is concluded with a brief examination of some specific EAs that are of experimental and theoretical Interest [250]. A variation of Gaussian-3 (G3) theory is presented in which the geometries and zero-point energies are obtained from B3LYP DFT [B3LYP/6-31G (d)] instead of geometries from second-order perturbation theory [MP2 (FU)/6-31G (d)] and zero-point energies from Hartree – Fock theory [HF/6-31G(d)]. This variation, referred to as G3//B3LYP, is assessed on 299 energies (enthalpies of formation, ionization potentials, electron affinities, proton affinities) from the G2/97 test set [251]. The G3//B3LYP average absolute deviation from experiment for the 299 energies is 0.99 kcal mol21 compared to 1.01 kcal mol21


for G3 theory. Generally, the results from the two methods are similar, with some exceptions. G3//B3LYP theory gives significantly improved results for several cases for which MP2 theory is deficient for optimized geometries, such as CN and Oþ 2 . However, G3//B3LYP does poorly for ionization potentials that involve a Jahn –Teller distortion þ þ in the cation (CHþ 4 , BF3 , BCl3 ) because of the B3LYP/631G(d) geometries. The G3 (MP2) method is also modified to use B3LYP/6-31G(d) geometries and zero-point energies Thus, use of density functional geometries and zero-point energies in G3 and G3(MP2) theories is a useful alternative to MP2 geometries and HF zero-point energies [252]. Effect of electron correlation on geometries is taken into account at MP2 and density functional approach at B3LYP level. Energy calculations at different levels up to CISD/631þ þ G (d,p)//MP2/6-31 þ G3* are reported for accurate estimation of conformational stability [253]. 8.4. Relative energies between isomers The low-energy and high-energy conformers of the three isomers of 1, 4-difluorobutadiene (DFBD) were studied with the Gaussian-3 (G3) and G3//B3LYP (G3B3) methods. The geometrical structures of the conformers were gradient optimized by the HF and B3LYP methods with the 6-31G (d,p) basis set. Natural bond orbital (NBO) analysis also was performed at the same levels of theory. The computational results show that the high-energy and low-energy conformers of a given isomer is essentially the same in geometry, except that their torsional angles about the C – C bonds are different, and among them only the high-energy conformer of the cis – trans isomer is coplanar [254]. Ab-initio calculations were used to identify the stable rotamers of N-methyl-2-fluoroacetamide (NMFA) and N-methyl-2-fluoropropionamide (NMFP). And obtain their geometries and the application of solvation theory on the 1JCF coupling constant gave the conformer populations in the solvents. In NMFA ab-initio calculations at the CBS-Q level yielded only two stable rotamers, the cis and trans, with DEðcis – transÞ ¼ 19:7 kJ mol21 : The presence of two conformers was confirmed by the FTIR spectra. Assuming these forms, the observed couplings when analyzed by solvation theory gave DE ¼ 21:3 kJ mol21 in the vapor phase, decreasing to 8.9 kJ mol21 in CDCl3 and to 0.8 kJ mol21 in DMSO. For NMFP the B3LYP calculations at the 6-311þ þ G (2df,2p) level gave only the trans rotamer as stable, while the gauche form was a plateau in the PES [255]. The geometry of various tautomers and isomers of 2-amino-2-imidazoline, 2-amino-2-oxazoline and 2-amino-2-thiazoline have been studied using B3LYP/6311 þ G (d,p) DFT, MP2/6-311 þ G(d,p) and CBS-Q model. At each of the foregoing levels of theory,

13

the amino tautomer is computed to be more stable than the imino tautomer [256]. Spectroscopic characterization and single crystal structure determination results are in good agreement with G3 (MP2) ab-initio calculations for the strain energy of the parent CPSiH5 compound [257].

9. Entropy contributions, free energy and enthalpy Absolute pK as of the hydroxyquinolines in aqueous solution are calculated through application of high level MO gas-phase Gibbs energies and solvation Gibbs energies using a variety of different methods. The gas-phase energies were obtained at the G3 (MP2) level, while solvation energies were calculated using six different methods; the Langevin dipole (LD) and Poisson – Boltzmann (PB) methods, two semi-empirical quantum mechanical methods SM2 and SM5.42R/A, and the polarizable continuum models PCM and IEF-PCM [258]. By using the Gaussian3 (G3) and Gaussian-2 (G2) methods the heats of formation of a range of chlorofluoro-, bromofluoro-, and iodofluoromethanes, methyls, and carbenes were computed. Also the vertical excitation energies of CF2Cl2 and CF2Br2 (to the lowest singlet and triplet B1, A2, B2, and A1 excited states) were computed using a range of methods including CASPT2 and EOM-CCSD in conjunction with the cc-pVTZ basis set [259]. Heats of formation have been calculated at the G2 (MP2) and G2 levels through both atomization reactions and bond separation isodesmic reactions for a series of saturated and unsaturated alicyclic hydrocarbons varying the size and the number of formal double bonds in the molecule [260]. Gaussian-3 (G3) theory was shown to be an appropriate method for studying glyoxal unimolecular dissociation since the calculated heats of reaction agree well with reliable experimental values. On the other hand, other standard model chemistries, including the popular HDFT methods predict poor heats of reaction for glyoxal dissociation [261]. Heats of formation for AlH, AlOH, OAlH, and OAlOH molecules and their monocations are determined at the B3LYP, B3PW91, mPW1PW91 and B1LYP levels of DFT in conjunction with a series of extended basis sets, and at the G3, G3B3, CBS-Q and CBS-QB3 levels of the ab initio MO theory [262]. Ab-initio calculations at the CBS-Q level, with additional HF/6-31G(d0 ) PES is applied to define the hindrance potential for internal rotations, heats of formation, entropies, and heat capacity values ½Cp ðTÞ of species involved in prototypical H abstraction reactions [263]. Enthalpy, entropy, and heat capacities, Cp ðTÞ; from 300 to 1500 K are determined for three chlorodimethyl ethers by density functional and ab-initio calculation methods. Molecular structures and vibration frequencies are determined at the B3LYP/6-31G (d,p) density functional calculation level, with single point calculations for energy at the B3LYP/6-311 þ G(3df,2p),

14


QCISD(T)/6-31G(d,p), and CBS-Q//B3LYP/6-31G(d,p) levels of calculation [264]. Enthalpy, entropy, and heat capacities, Cp ðTÞ from 300 to 1500 K are determined for Me-hypochlorite and three chloromethyl hypochlorites by density functional and ab-initio calculation methods. Molecular structures and vibration frequencies are determined at the B3LYP/6-31G(d,p) density functional calculation level, with single-point calculations for energy at the B3LYP/6-31G(d,p), B3LYP/6-311 þ G(3df,2p), QCISD(T)/6-31G(d,p), and CBS-Q levels of calculation [265]. Gaussian-3 (G3) method with B3LYP/6-31G (d) and MP2/cc-pVTZ geometries (denoted as G3B3 and G3//MP2 here, respectively) is used for calculating enthalpies of formation and the atomization procedure using both G3B3 and G3//MP2 predicts the enthalpies of formation larger than the experimental [266]. Two computational procedures, the density functional B3PW91/6-311 þ G(2df,2pd) and the ab-initio CBS-Q, have been used to calculation optimized geometries, energy minimum at 0 K and enthalpies and free energies at 298 K for atoms and molecules involved in the ignition and/or combustion of boron [267]. The enthalpies of formation were determined for 14 intermediate species of cyclopentadiene oxidation under combustion conditions by ab-initio MO calculations at the G2 (MP2, SVP) and G2 (B3LYP/MP2, SVP) levels of theory and with the use of isodesmic reactions. The G2 (B3LYP/MP2, SVP) method, a hybrid of G2 (MP2, SVP) and the density function-based G2 (B3LYP/MP2), was devised recently [268]. The atomization energies of the 55 G2 molecules are computed using the B3LYP approach with a variety of basis sets. The 6-311 þ G (3df) basis set is found to yield superior results to those obtained using the augmented-correlation-consistent valence-polarized triplezeta set [269]. Singlet – triplet separation and heat of formation of phenylcarbene were calculated by using the B3LYP, RCCSD (T), G2M, G3, CASSCF, CASPT2 and MRCI methods [270]. The heats of formation of haloacetylenes are evaluated using the recent W1 and W2 ab initio computational thermochemical methods. These calculations involve CCSD and CCSD (T) coupled cluster methods, basis sets of up to spdfgh quality, extrapolations to the one-particle basis set limit, and contributions of inner-shell correlation, scalar relativistic effects, and (relevant) firstorder spin-orbit coupling. The performance of composite thermochemical schemes such as G2, G3, G3X and CBS-QB3 theories was analyzed. Of the other methods, CBS-RAD and G3 (MP2)-RAD produce good BDEs. A cancellation of errors leads to reasonable RSEs being produced from all the methods examined CBS-RAD, W10 and G3 (MP2)-RAD perform best, while UB3-LYP performs worst [271]. The ground-state PES, is studied employing the B3LYP/6-311G (d,p) and RCCSD(T)/6311 þ G(3df,2p) levels of theory. The first excited doublet state PES, is also studied by utilizing the CASSCF

(11,11)/6-311 þ G (d,p) and MRCI þ D(7,8)/ANO(2 þ ) levels of theory. Thermochemical parameters for the C3Hn ðn ¼ 1 – 4Þ species are determined by employing the G3 theory and the CCSD (T)/6-311 þ G (3df, 2p) method [272]. Thermochemical properties are computed by density functional B3LYP/6-31G(d,p) and B3LYP/6311 þ G(3df,2p), ab-initio QCISD(T)/6-31G(d,p), and composite CBS-Q calculation methods for chlorinated aldehydes and the corresponding chlorinated acetyl and formyl Me radicals [273]. Ab-initio and DFT calculations are reported for the chlorofluoroamines HNXY (X/Y ¼ F/ Cl) and all possible unimolecular reaction products from their ground state. Reliable enthalpies of formation for these molecules and reaction products have been calculated using the G2 model. Optimized structures of all the species have been obtained at various levels up to MP2/6-311þ þ G** and B3LYP/6-311þ þ G**. The triplet-singlet energy gaps have been estimated for NH, NF and NCl at various levels. Enthalpies of various reactions have been calculated at advanced levels including PMP4, CBS-Q, G1, G2, and G2-MP2 [274]. Ab-initio and DFT studies were performed on three isomers of tetradehydrobenzene (benzdiynes). Four different density functionals (BPW91, BLYP, B3LYP, and B1LYP) and two higher levels of theory [QCISD and CCSD (T)] incorporating basis sets up to Dunning’s correlationconsistent polarized valence triple-j (cc-pVTZ) were utilized for this purpose. Stability tests showed that more stable solutions were available for 1,4-benzdiyne with unrestricted than with restricted DFT, while solutions obtained with later descriptions of 1,3-benzdiyne and 1,2,3,5-tetradehydrobenzene were stable. UB3LYP provided better geometry for 1,4-benzdiyne. The heats of formation of benzdiynes were obtained by using the G2, CBS-Q, and CBS-QB3 methodologies. The heats of formation calculated for 1,3-benzdiyne and 1,2, 3,5-tetradehydrobenzene were 208.6 and 197.9 kcal mol21, respectively, at the CBS-QB3 level of theory [275]. Thermochemical properties and reaction path parameters of reactions of allylic isobutenyl radical (C – C(C) – C) with molecular O are determined by ab-initio-Moller –Plesset (MP2(full)/6-31G(d) and MP4(full)/6-31G(d,p)//MP2(full)/ 6-31G(d)), complete basis set model chem. (CBS-4 and CBS-q with MP2(full)/6-31G(d) and B3LYP/6-31G(d) optimized geometries), and d. functional (B3LYP/6-31G(d) and B3LYP/6-311 þ G(3df,2p)//B3LYP/6-31g(d)) calculations [276]. Thermochemical properties, enthalpy, entropy, and heat capacity (300 – 1500 K), are computed by density functional B3LYP/6-31G (d,p) and B3LYP/6-311 þ G(3df,2p), ab initio QCISD(T)/6-31G(d0 ), and composite CBS-Q calculation methods for seven chlorovinyl alcohols.: CH2: COHCl, (E)-CHCl:CHOH, (Z)-CHCl:CHOH, CCl2:CHOH, (E)-CHCl:COHCl, (Z)-CHCl:COHCl, and CCl2:CClOH. Molecular structures and vibration frequencies are


determined at the B3LYP/6-31G (d,p) level of theory. Also using the rigid-rotor-harmonic-oscillator approximation based on the B3LYP/6-31G (d,p) structures hindered internal rotational contributions to entropies and heat capacities are calculated [277]. Thermochemical and reaction mechanism of tetrazole (H2CN4) and tetrazolate anion (HCN2 4 ) are studied by ab-initio methods including MP2/6-31G**, B3LYP/631G**, B3LYP/6-311 þ G (2d,p), and CBS/QB3 from GAUSSIAN 98 program [278]. The thermodynamic properties for reactants, intermediates, TSs, and products of the methyl tert-butyl ether pyrolysis, the potential energy diagram for MTBE unimolecular elimination and oxidation reactions are calculated from CBS-Q//B3LYP/6-2lG(d) [279].Thermochemical parameters of tert-Bu radical are determined by ab-initio-Moller –Plesset (MP2 (full)/6-31G(d)), complete basis set model chememisteries (CBS-4 and CBS-Q with MP2(full)/6-31G(d) and B3LYP/6-31G(d) optimized geometries), density functional (B3LYP/6-31G(d)), semiempirical MOPAC (PM3) MO calculations, and by group additivity estimation [280]. The heats of formation of four C2H3N molecules and the three most stable C2H2N cations, namely, cyanomethyl, isocyanomethyl, and azirinyl, together with their corresponding radicals and anions, have been calculated at the G2, G2(//QCI), CBS-Q, and CBS-RAD levels of theory. In addition, bond dissociation enthalpies and gas-phase acidities of the molecules and the electron affinities and ionization energies of the free radicals have been derived [281]. Finally, for reactions of the HSCH2 radical with atmosphere gases O2, NO, and NO2, the thermochemistry is described by HF, MP2, and density functional computational methods, by Petersson’s complete basis set extrapolation (CBS-4 and CBS-Q), and, in part, by Pople’s G2MP2 model chemistry method [282].

10. Recommendations for obtaining a particular set of data Gaussian-based computations can be categorized as geometry optimization and property calculation. Geometry optimization is a fundamental component of molecular modeling. The determination of a low-energy conformation for a given force field can be the final objective of the computation. Alternatively, the minimum for the system on the specified PES, in a local or global sense, can serve as a starting or reference point for subsequent calculations. All the methods in Section 2 can be used for this propose. It is clear that choosing the optimization method and level of calculation depends on what we aim at. For example for calculating the transition state structures the precision and accuracy of the calculation is vital. But for studying the trend of the bond length change in the series of compound just the accuracy is important. Distance matrix of optimized geometry is powerful tool for checking the final structure.

15

The concept of a PES is centeral to the discussion of molecular structure and geometry optimization. The PES describes the energy of the molecule as a function of its geometry (i.e. bond lengths, valance angels, torsions and other internal coordinates). Equilibrium geometries are local minima on the PES for a molecule; transition states correspond to saddle points on the PES. Geometry optimization is the process of finding these minima and saddle points. PES arise naturally from the BO approximation. Because the electrons are so much lighter than the nuclei, the electronic part of the wave-function can read just almost instantaneously to any nuclear motion. In the BO approximation, a PES is obtained by solving for the electronic energy at a series of fixed nuclear positions. This is usually quite satisfactory for most ground state systems. However, for photochemical systems, which involve excited state surfaces as well as the ground state, one must go beyond the BO approximation to treat molecular motion near seam, conical intersections and weakly avoided crossings. PESs make it possible to discuss molecular structures. The equilibrium geometry of a molecule corresponds to a minimum on the PES. There may be several minima, representing different conformers and isomers of the molecule, or representing reactants, intermediates and products of a chemical reaction. If the valley on the PES is deep with steep side, then the structure of the molecule is fairly rigid and well defined. However, if the valley is broad or shallow, the molecule is flexible or reactive, and the concept of molecular structure is less well defined. Since accurate functions for PES are difficult to obtain even for very small systems, most geometry optimization methods find equilibrium structures and transition structures directly, without constructing the full PES. Efficient geometry optimization methods employ the first derivatives of the PESs with respect to the geometric parameters. Usually these are obtained analytically, but for theoretical methods without analytical derivatives they can be calculated numerically. For some algorithms, second derivatives can also be used. The first derivatives of the PES are also called the gradient. In classical mechanics, the negatives of the first derivatives of the potential are the forces on the atoms in the molecule, Thus points on the PES where the gradient or forces are zero are called stationary points. In a topological analysis of the PESs, these points are known as critical points. The matrix of second derivatives of the PES is termed the Hessian matrix or the harmonic force constant matrix. At a critical point, diagonalization of the massweighted force constant matrix yields the vibrational frequencies and normal modes. The number of negative eigenvalues of the Hessian or the number of imaginary frequencies at a stationary or critical point is known as the index of the critical point. A critical point of index 0 is minimum and a critical point of index 1 is a transition structure.

16


For a point on the PES to be a minimum, it must satisfy two conditions. The first derivatives, or equivalently the gradient or the forces must be zero. If the first derivatives are not zero, there is a nearby point that is lower in energy. Secondly, the second derivative matrix or Hessian or force constant matrix must be positive definite. In other words, all the eigenvalues of the Hessian must be positive for a minimum or all of the vibrational frequencies must be real (no imaginary frequency). If one or more eigenvalues are negative, then the potential surface is a maximum along these directions and the point is not a minimum but a saddle point. Full optimization of molecular structures has been carried out routinely with the Hartree – Fock ab-initio method since the 1970s. It is known that the Hartree – Fock method predicts bond lengths that are too short, while MP2 or other correlated level of ab-initio methods predict bond lengths that are too long, especially for multiple bonds. The highly correlated CCSD (T) method using a large basis set yields very accurate computed geometries. However, the approach can be applied to small molecules only. Being much more Cost-Effective computationally, DFT offers an alternative for the prediction of accurate molecular geometries for larger systems. To day, quantum chemists aim at the development of different methods in order to get accurate results. Some of these developments are given in the following. A systematic method [283] for the determination of exchange-correlation functionals within the generalized gradient approximation (GGA) is applied to extended G2 test set of standard heats of formation of Curtiss et al. [284].Recently a similar methodology that goes beyond the GGA,by taking second-order gradients and the (noninteracting) kinetic-energy density into account, is reported [285]. A recently suggested procedure for the systematic optimization of gradient-correction exchange-correlation functionals [286] has been applied to the extended G2 test set [287], which consists of the standard heats of formation of 148 molecules [288]. The performance of G2 (MP2) and G2 (MP2, SVP) theories for molecules containing third-row non-transition elements Ga –Kr is assessed. The average absolute deviation from experimental for 40 test energies is 1.92 kcal mol 21 for both methods compared to 1.37 kcal mol21mol for G2 theory The B3PW91 DFT gives the best agreement with experimental in contrast to first and second-row systems, where B3LYP does better than B3PW91 [289]. A comparison of several methods for calculating gas-phase acidities is reported. Methods examined include DFT with the B-LYP, B-P86, B3-LYP and B3-P86 non-local functionals, second-order [MP2] and fourth-order [MP4] Moeller-Plesset and fourth-order Feenberg [F4] theories, quadratic CI (QCISD(T)), and variants of G2 theory including G2(MP2, SVP), G2(MP2) and G2 itself. Excellent results are obtained with the G2 (MP2, SVP), G2 (MP2) and G2 procedures. MP4 and F4 methods using the 6-311 þ G (3df, 2p) basis produce similar

excellent agreement with experimental [290]. The performances of a variety of procedures were assessed for calculating gas-phase proton affinities. Methods examined include DFT with the B-LYP and Becke3-LYP non-local functionals, second-order (MP2) and fourth-order (MP4) Moeller– Plesset and fourth-order Feenberg (F4) theories, quadratic CI (QCISD(T)), G2(MP2) (G2 ¼ Gaussian-2), and G2. The MP4, F4 and QCISD (T) proton affinities when calculated. With the 6-311G (d, p) basis set were not in good agreement with the available experimental data, largely reflecting basis-set inadequacies. On the other hand, excellent results were obtained with these procedures and with G2 (MP2) and G2, when the 6-311 þ G (3df, 2p) basis set was used [291]. The thermal 1,3-sigmatropic migrations in X –CH2 –CH:CH2 with X ¼ BH2, NH2, and Me were studied at the RHF, MP2, B3LYP, G3, and CBS-APNO levels using basis sets with added diffuse and polarization functions. In all three cases, the suprafacial allowed path proceeds through a TS in which the C:C electron is delocalized into the 2p AO in the migrating X [292]. Bond dissociation energies (BDEs) and radical stabilization energies (RSEs) associated with 22 monosubstituted Me radicals (–bul –CH2X) were determined at a variety of levels including, CBS-RAD, G3 (MP2)-RAD, RMP2, UB3-LYP and RB3-LYP. In addition, W10 values were obtained for a subset of 13 of the radicals. The W10 BDEs and RSEs are generally close to experimental [293]. The heats of formation of TNAZ (1,3,3-trinitroazetidine) and related compounds have been calculated with the G3(MP2)//B3LYP model to give a mean abs. error of 0.85 kcal mol21 for 16 singlet species and 0.61 kcal mol21 for 12 free radical intermediates. These uncorrected errors are better than our previous B3LYP/6-31G (d,p)-based scheme that employed seven auxiliary parameters. The MAD of the G3 (MP2)//B3LYP energies of the combined set of 28 singlet and radical species, 0.75 kcal mol21, can be reduced to 0.42 kcal mol21 when supplemented with a minimal four-parameter atom-based correction scheme. It was found that the previous B3LYP/631G (d,p) energies when fitted to a combination of 39 experimental and G3(MP2)//B3LYP singlet and radical species gave a mean absolute deviation of 3.7 kcal mol21, but that the error is reduced to 1.52 kcal mol21 when the B3LYP/6-31G(d,p) energies are supplemented with a five parameter additive correction scheme. The results of these calculations are used to clarify the initial mechanistic steps in the decomposition of TNAZ [294]. Energies were calculated at B3LYP, MP2 and CCSD (T) levels of theory, and in a G2 (MP2)-like manner. The final platinum basis set, ‘LANL2TZ þ (3f2g)’,was combined with the G3MP2 large basis set on the ligands to yield a final G3 (MP2)// B3LYP-like energy [295]. The performance of Gaussian-3 (G3) theory and six related methods for the calculation of enthalpies of formation of n-alkanes of up to 16 carbons and isoalkanes of up to 10 carbons. Also the accuracy of the B3LYP DFT for the n-alkanes have been examined. The G3


enthalpies of formation of the n-alkanes have errors of less than 2 kcal mol21 compared to experimental. The B3LYP method does very poorly for the calculation of enthalpies of formation of the larger n-alkanes with an error of over 30 kcal mol21 for hexadecane. This suggests that B3LYP has a significant problem with accumulation of errors as the molecular size increases. Several schemes for correcting systematic errors in B3LYP calculations for large molecules are also explored [296]. The G2/97 test set for assessing quantum chem. methods used to predict thermochemical data is expanded to include 75 additional enthalpies of formation of larger molecules. This new set, referred to as the G3/99 test set, includes enthalpies of formation, ionization potentials, electron affinities, and proton affinities in the G2/97 set and 75 new enthalpies of formation. The total number of energies in the G3/99 set is 376 [297]. An attempt has been made to use modern quantum methods to codify the data base concerning bond dissociation energies in hydrocarbons. Calculations have been performed using two hybrid DFT methods, the well-known B3LYP formalism and a newly developed alternative named KMLYP. CBS-Q method has also been employed [298]. The basis set convergence of non-local density functional theory (NL-DFT) using the functionals BLYP, BP86, B3P86, B3PW91 and B3LYP has been investigated. To this the heats of hydrogenation of N2, N2H2 and N2H4 at the NL-DFT levels using the Pople basis sets from 6-31G(d) up to 6-311 þ G(3df,3pd), Dunnings correlation consistent basis sets cc-pVDZ, cc-pVTZ, cc-pVQZ and their augmented versions, are reported. Also additional calculations at the BLYP level using Slater-type basis functions up to TZ2P quality are done [299]. Extensive complete basis set ab-initio and DFT computational studies of selected chemical systems were performed in order to evaluate the most accurate computational method for the evaluation of their proton affinities. Three groups hybrid, gradient-correction, and local DFT methods in combination with the 6-311G (2d,2p) basis set were used for these studies [300]. The recently introduced complete basis set, CBS-Q, model chememisteries is modified to use B3LYP hybrid density functional geometries and frequencies, which give both improved reliability (max. error for the G2 test set reduced from 3.9 to 2.8 kcal mol21) and increased accuracy (mean absolute error reduced from 0.98 to 0.87 kcal mol21), with little penalty in computational speed [301]. The results of density functional (B3LYP) computations with a variety of basis sets were compared with MP2, MP4, QCISD (T), and CBS-Q model chemistries results and benchmarked against experimental data. The results show that geometrical features are already accurately described using B3LYP/6-311G (d,p) or MP2/6-311G(d,p). For a systematic study of the energetics of substitution on alkyl cations, B3LYP/6-311 þ G (d,p) and MP2/6-311 þ G(d,p)

17

computations form a useful compromise between accuracy (average deviation within 1 kcal mol21 of the experimental error) and computational efficiency. The electronic structures of these species and their precursors CH3CH2R were studied using both NBO and atoms-in-molecules (AIM) analyses [302]. Several high-accuracy Gaussian-2, complete basis set and density functional methods for computational thermochemistry (in order of increasing speed): G2, G2(MP2), CBS-Q, G2(MP2,SVP), CBS-Q, CBS-4, and B3LYP/6-311 þ G(3df,2p) are examined [303]. CBS-4, CBS-q, CBS-Q, and CBS-QCI/APNO models employ small basis set/low-order calculations to determine the molecular geometry and vibrational zero-point energy (e.g. CBS-4//HF/3-21G*). These methods then determined the electronic energy using large basis sets (with extrapolation to the CBS limit) and high-orders of correlation energy. Recently these models have been extended to transition states for chememical reactions by searching the ‘inexpensive’ (e.g. HF/3-21G*) intrinsic reaction coordinate (IRC) for the maximum energy of the high level calculation (e.g. CBS-4) [304].

11. Concluding remarks: the need for further development The ultimate goal of most quantum chemical approaches is the—approximate—solution of the timeindependent, non-relativistic Schro¨dinger equation. Although these approximations led to development the computational methods, these methods have disadvantages, which directly are related to these approximations. At least a correction term may be considered. The time dependent methods such as multiconfiguration time dependent Hatree –Fock method is rather new, and there are still several aspects that have to be improved. Potential representation and refining the numerical integrators for solving the non-linear multiconfiguration time dependent equations of motion are active fields. Perhaps the above requirements for better approximations to these methods are important however, in the near future, different avenues can be easily implemented. We believe that some modifications will more than likely be developed, due to increased capabilities of computational resources. It may be needed at times to use coupled-cluster geometries in GX (X ¼ 1, 2, 3, 3B3) as well as CBS-Q (CBS-QB3) methods. For small molecules or peroxides a modest basis set can be used in conjunction to CCSD geometries (and perhaps MP2 or QCISD frequencies) to yield more accurate energies. Also, the fitting parameterizations used in these methods should be updated to include the enormous amount of data on molecular structure that is currently available, to further reduce error. These are among some of the current areas of our research efforts [305].

18


12. Further Reading [54].

Acknowledgements We are very grateful for the useful suggestions, comments and encouragement that Professor Imre Csizmadia has provided us during the completion of this review article.

References [1] F. Jensen, Introduction to Computational Chemistry, Wiley, NewYork, 1999. [2] J.C. Slater, Phys. Rev. 36 (1930) 57. [3] S.F. Boys, Proc. R. Soc. (Lond.) A 200 (1950) 542. [4] W.J. Hehre, R.F. Stewart, J.A. Pople, J. Chem. Phys. 51 (1969) 2657. [5] J.B. Collins, P.v.R. Schleyer, J.S. Binkley, J.A. Pople, J. Chem. Phys. 64 (1976) 5142. [6] J.S. Binkley, J.A. Pople, W.J. Hehre, J. Am. Chem. Soc. 102 (1980) 939. [7] M.S. Gordon, J.S. Binkley, J.A. Pople, W.J. Pietro, W.J. Hehre, J. Am. Chem. Soc. 104 (1982) 2797. [8] W.J. Pietro, M.M. Francl, W.J. Hehre, D.J. Defrees, J.A. Pople, J.S. Binkley, J. Am. Chem. Soc. 104 (1982) 5039. [9] K.D. Dobbs, W.J. Hehre, J. Comp. Chem. 7 (1986) 359. K.D. Dobbs, W.J. Hehre, J. Comput. Chem. 8 (1987) 861. [10] K.D. Dobbs, W.J. Hehre, J. Comp. Chem. 8 (1987) 880. [11] R. Ditchfield, W.J. Hehre, J.A. Pople, J. Chem. Phys. 54 (1971) 724. [12] W.J. Hehre, R. Ditchfield, J.A. Pople, J. Chem. Phys. 56 (1972) 2257. [13] P.C. Hariharan, J.A. Pople, Mol. Phys. 27 (1974) 209. [14] M.S. Gordon, Chem. Phys. Lett. 76 (1980) 163. [15] P.C. Hariharan, J.A. Pople, Theor. Chim. Acta 28 (1973) 213. [16] T.H. Dunning Jr., P.J. Hay, in: H.F. Schaefer III (Ed.), Modern Theoretical Chemistry, vol. 3, Plenum Press, New York, 1976, p. 1. [17] A.K. Rappe, T. Smedly, W.A. Goddard III, J. Phys. Chem. 85 (1981) 1662. [18] W. Stevens, H. Basch, J. Krauss, J. Chem. Phys. 81 (1984) 6026. [19] W.J. Stevens, M. Krauss, H. Basch, P.G. Jasien, Can. J. Chem. 70 (1992) 612. [20] T.R. Cundari, W.J. Stevens, J. Chem. Phys. 98 (1993) 5555. [21] W.J. Hehre, R.F. Stewart, J.A. Pople, J. Chem. Phys. 51 (1969) 2657. [22] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 270. [23] W.R. Wadt, P.J. Hay, J. Chem. Phys. 82 (1985) 284. [24] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 299. [25] P. Fuentealba, H. Preuss, H. Stoll, L.v. Szentpaly, Chem. Phys. Lett. 89 (1989) 418. [26] L.v. Szentpaly, P. Fuentealba, H. Preuss, H. Stoll, Chem. Phys. Lett. 93 (1982) 555. [27] P. Fuentealba, H. Stoll, L.v. Szentpaly, P. Schwerdtfeger, H. Preuss, J. Phys. B16 (1983) 1323. [28] H. Stoll, P. Fuentealba, P. Schwerdtfeger, J. Flad, L.v. Szentpaly, H. Preuss, J. Chem. Phys. 81 (1984) 2732. [29] P. Fuentealba, L.v. Szentpaly, H. Preuss, H. Stoll, J. Phys. B 18 (1985) 1287. [30] I.N. Levine, Quantum Chemistry, fifth ed., Prentice Hall, New Jersey, 2000. [31] M. Dolg, U. Wedig, H. Stoll, H. Preuss, J. Chem. Phys. 86 (1987) 866. [32] G. Igel-Mann, H. Stoll, H. Preuss, Mol. Phys. 65 (1988) 1321. [33] M. Dolg, H. Stoll, H. Preuss, J. Chem. Phys. 90 (1989) 1730.

[34] P. Schwerdtfeger, M. Dolg, W.H.E. Schwarz, G.A. Bowmaker, P.D.W. Boyd, J. Chem. Phys. 91 (1989) 1762. [35] M. Dolg, H. Stoll, A. Savin, H. Preuss, Theor. Chim. Acta 75 (1989) 173. [36] D. Andrae, U. Haeussermann, M. Dolg, H. Stoll, H. Preuss, Theor. Chim. Acta 77 (1990) 123. [37] M. Kaupp, P.v.R. Schleyer, H. Stoll, H. Preuss, J. Chem. Phys. 94 (1991) 1360. [38] W. Kuechle, M. Dolg, H. Stoll, H. Preuss, Mol. Phys. 74 (1991) 1245. [39] M. Dolg, P. Fulde, W. Kuechle, C.-S. Neumann, H. Stoll, J. Chem. Phys. 94 (1991) 3011. [40] M. Dolg, H. Stoll, H.J. Flad, H. Preuss, J. Chem. Phys. 97 (1992) 1162. [41] A. Bergner, M. Dolg, W. Kuechle, H. Stoll, H. Preuss, Mol. Phys. 80 (1993) 1431. [42] M. Dolg, H. Stoll, H. Preuss, Theor. Chim. Acta 85 (1993) 441. [43] M. Dolg, H. Stoll, H. Preuss, R.M. Pitzer, J. Phys. Chem. 97 (1993) 5852. [44] U. Haeussermann, M. Dolg, H. Stoll, H. Preuss, Mol. Phys. 78 (1993) 1211. [45] W. Kuechle, M. Dolg, H. Stoll, H. Preuss, J. Chem. Phys. 100 (1994) 7535. [46] A. Nicklass, M. Dolg, H. Stoll, H. Preuss, J. Chem. Phys. 102 (1995) 8942. [47] T. Leininger, A. Nicklass, H. Stoll, M. Dolg, P. Schwerdtfeger, J. Chem. Phys. 105 (1996) 1052. [48] D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 98 (1993) 1358. [49] R.A. Kendall, T.H. Dunning Jr., R.J. Harrison, J. Chem. Phys. 96 (1992) 6796. [50] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007. [51] K.A. Peterson, D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 100 (1994) 7410. [52] A. Wilson, T. van Mourik, T.H. Dunning Jr., Theochem 388 (1997) 339. [53] E.R. Davidson, Chem. Phys. Lett. 220 (1996) 514. [54] C.C.J. Roothan, Rev. Mod. Phys. 23 (1951) 69. [55] J.A. Pople, R.K. Nesbet, J. Chem. Phys. 22 (1954) 571. [56] R. McWeeny, G. Dierksen, J. Chem. Phys. 49 (1968) 4852. [57] M. Head-Gordon, J.A. Pople, M.J. Frisch, Chem. Phys. Lett. 153 (1988) 503. [58] M.J. Frisch, M. Head-Gordon, J.A. Pople, Chem. Phys. Lett. 166 (1990) 275. [59] M.J. Frisch, M. Head-Gordon, J.A. Pople, Chem. Phys. Lett. 166 (1990) 281. [60] M. Head-Gordon, T. Head-Gordon, Chem. Phys. Lett. 220 (1994) 122. [61] S. Saebo, J. Almlof, Chem. Phys. Lett. 154 (1989) 83. [62] C. Moller, M.S. Plesset, Phys. Rev. 46 (1934) 618. [63] J.A. Pople, R. Seeger, R. Krishnan, Int. J. Quant. Chem. Symp. 11 (1977) 149. [64] J.A. Pople, J.S. Binkley, R. Seeger, Int. J. Quant. Chem. Symp. 10 (1976) 1. [65] R. Krishnan, J.A. Pople, Int. J. Quant. Chem. 14 (1978) 91. [66] K. Raghavachari, J.A. Pople, E.S. Replogle, M. Head-Gordon, J. Phys. Chem. 94 (1990) 5579. [67] J.A. Pople, R. Krishnan, H.B. Schlegel, J.S. Binkley, Int. J. Quant. Chem., Quant. Chem.Symp. 13 (1979) 325. [68] N.C. Handy, H.F. Schaefer III, J. Chem. Phys. 81 (1984) 5031. [69] G.W. Trucks, E.A. Salter, C. Sosa, R.J. Bartlett, Chem. Phys. Lett. 147 (1988) 359. [70] G.W. Trucks, J.D. Watts, E.A. Salter, R.J. Bartlett, Chem. Phys. Lett. 153 (1988) 490. [71] J.B. Foresman, M. Head-Gordon, J.A. Pople, M.J. Frisch, J. Phys. Chem. 96 (1992) 135. [72] J.B. Foresman, Æ. Frisch, Exploring Chemistry with Electronic Structure Methods, second ed., Gaussian, Inc, Pittsburgh, PA, 1996.

A.F. Jalbout et al. / Journal of Molecular Structure (Theochem) 671 (2004) 1–21 [73] R. Krishnan, H.B. Schlegel, J.A. Pople, J. Chem. Phys. 72 (1980) 4654. [74] K. Raghavachari, J.A. Pople, Int. J. Quant. Chem. 20 (1981) 167. [75] D. Hegarty, M.A. Robb, Mol. Phys. 38 (1979) 1795. [76] R.H.E. Eade, M.A. Robb, Chem. Phys. Lett. 83 (1981) 362. [77] H.B. Schlegel, M.A. Robb, Chem. Phys. Lett. 93 (1982) 43. [78] F. Bernardi, A. Bottini, J.J.W. McDougall, M.A. Robb, H.B. Schlegel, Far. Symp. Chem. Soc. 19 (1984) 137. [79] N. Yamamoto, T. Vreven, M.A. Robb, M.J. Frisch, H.B. Schlegel, Chem. Phys. Lett. 250 (1996) 373. [80] M.J. Frisch, I.N. Ragazos, M.A. Robb, H.B. Schlegel, Chem. Phys. Lett. 189 (1992) 24. [81] J.A. Pople, R. Krishnan, H.B. Schlegel, J.S. Binkley, Int. J. Quant. Chem. XIV (1978) 545. [82] R.J. Bartlett, G.D. Purvis, Int. J. Quant. Chem. 14 (1978) 516. [83] J. Cizek, Adv. Chem. Phys. 14 (1969) 35. [84] G.D. Purvis, R.J. Bartlett, J. Chem. Phys. 76 (1982) 1910. [85] G.E. Scuseria, C.L. Janssen, H.F. Schaefer III, J. Chem. Phys. 89 (1988) 7382. [86] G.E. Scuseria, H.F. Schaefer III, J. Chem. Phys. 90 (1989) 3700. [87] K. Raghavachari, J.A. Pople, E.S. Replogle, M. Head-Gordon, J. Phys. Chem. 94 (1990) 5579. [88] J. Gauss, C. Cremer, Chem. Phys. Lett. 150 (1988) 280. [89] E.A. Salter, G.W. Trucks, R.J. Bartlett, J. Chem. Phys. 90 (1989) 1752. [90] N.C. Handy, J.A. Pople, M. Head-Gordon, K. Raghavachari, G.W. Trucks, Chem. Phys. Lett. 164 (1989) 85. [91] C.E. Dykstra, Chem. Phys. Lett. 45 (1977) 466. [92] J.V. Ortiz, J. Chem. Phys. 89 (1988) 6348. [93] L.S. Cederbaum, J. Phys. B8 (1975) 290. [94] W. von Niessen, J. Schirmer, L.S. Cederbaum, Comp. Phys. Rep. 1 (1984) 57. [95] V.G. Zakrzewski, W. von Niessen, J. Comp. Chem. 14 (1993) 13. [96] V.G. Zakrzewski, J.V. Ortiz, Int. J. Quant. Chem., Quant. Chem. Symp. 28 (1994) 23 –27. [97] J.V. Ortiz, Int. J. Quant. Chem., Quant. Chem. Symp. 22 (1988) 431. [98] J.V. Ortiz, Int. J. Quant. Chem., Quant. Chem. Symp. 23 (1989) 321. [99] J.V. Ortiz, V.G. Zakrzewski, O. Dolgounitcheva, One-electron pictures of electronic structure: propagator calculations on photoelectron spectra of aromatic molecules, in: J.-L. Calais, E.S. Kryachko (Eds.), Conceptual Perspectives in Quantum Chemistry, vol. 3, Kluwer, Dordrecht, 1997, pp. 465 –517. [100] P. Hohenberg, W. Kohn, Phys. Rev. 136B (1964) 864. [101] W. Kohn, L.J. Sham, Phys. Rev. 140A (1965) 1133. [102] D.R. Salahub, M.C. Zerner (Eds.), The Challenge of d and f Electrons, ACS, Washington, DC, 1989. [103] R.G. Parr, W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford University Press, Oxford, 1989. [104] B.G. Johnson, M.J. Frisch, J. Chem. Phys. 100 (1994) 7429. [105] B.G. Johnson, M.J. Frisch, Chem. Phys. Lett. 216 (1993) 133. [106] R.E. Stratmann, J.C. Burant, G.E. Scuseria, M.J. Frisch, J. Chem. Phys. 106 (1997) 10175. [107] J.A. Pople, M. Head-Gordon, D.J. Fox, K. Raghavachari, L.A. Curtiss, J. Chem. Phys. 90 (1989) 5622. [108] L.A. Curtiss, C. Jones, G.W. Trucks, K. Raghavachari, J.A. Pople, J. Chem. Phys. 93 (1990) 2537. [109] L.A. Curtiss, K. Raghavachari, G.W. Trucks, J.A. Pople, J. Chem. Phys. 94 (1991) 7221. [110] L.A. Curtiss, K. Raghavachari, J.A. Pople, J. Chem. Phys. 98 (1993) 1293. [111] J.A. Pople, Rev. Mod. Phys. 71 (5) (1999) 1267. [112] M.R. Nyden, G.A. Petersson, J. Chem. Phys. 75 (1981) 1843. [113] G.A. Petersson, M.A. Al-Laham, J. Chem. Phys. 94 (1991) 6081. [114] G.A. Petersson, T.G. Tensfeldt, J.A. Montgomery Jr., J. Chem. Phys. 94 (1991) 6091. [115] J.A. Montgomery Jr., J.W. Ochterski, G.A. Petersson, J. Chem. Phys. 101 (1994) 5900.

19

[116] J.W. Ochterski, G.A. Petersson, J.A. Montgomery Jr., J. Chem. Phys. 104 (1996) 2598. [117] G.A. Petersson, A. Bennett, T.G. Tensfeldt, M.A. Al-Laham, W.A. Shirley, J. Mantzaris, J. Chem. Phys. 89 (1988) 2193. [118] R. Bauernschmitt, R. Ahlrichs, Chem. Phys. Lett. 256 (1996) 454. [119] M.E. Casida, C. Jamorski, K.C. Casida, D.R. Salahub, J. Chem. Phys. 108 (1998) 4439. [120] M.A. Thompson, M.C. Zerner, J. Am. Chem. Soc. 113 (1991) 8210. [121] M.C. Zerner, Semi empirical molecular orbital methods, in: K.B. Lipkowitz, D.B. Boyd (Eds.), Reviews of Computational Chemistry, vol. 2, VCH Publishing, New York, 1991, p. 313. [122] M.C. Zerner, P. Correa de Mello, M. Hehenberger, Int. J. Quant. Chem. 21 (1982) 251. [123] L.K. Hanson, J. Fajer, M.A. Thompson, M.C. Zerner, J. Am. Chem. Soc. 109 (1987) 4728. [124] A.D. Bacon, M.C. Zerner, Theor. Chim. Acta 53 (1979) 21. [125] W.P. Anderson, W.D. Edwards, M.C. Zerner, Inorganic Chem. 25 (1986) 2728. [126] M.C. Zerner, G.H. Lowe, R.F. Kirchner, U.T. Mueller-Westerhoff, J. Am. Chem. Soc. 102 (1980) 589. [127] J.E. Ridley, M.C. Zerner, Theor. Chim. Acta 32 (1973) 111. [128] J.E. Ridley, M.C. Zerner, Theor. Chim. Acta 42 (1976) 223. [129] J.J.P. Stewart, Semiempirical Methods: Integrals and Scaling, Encyclopedia of Computational Chemistry, vol. 3, Wiley, New York, 2000, pp. 1614. [130] J. Segal, Pople, J. Chem. Phys. (1966) 3289. [131] J.A. Pople, D. Beveridge, P. Dobosh, J. Chem. Phys. 47 (1967) 2026. [132] M. Dewar, W. Thiel, J. Am. Chem. Soc. 99 (1977) 4499. [133] R.C. Bingham, M. Dewar, J. Am. Chem. Soc. 97 (1975) 1285. [134] M.J.S. Dewar, W. Thiel, J. Am. Chem. Soc. 99 (1977) 4899. [135] M.J.S. Dewar, H.S. Rzepa, J. Am. Chem. Soc. 100 (1978) 777. [136] M.J.S. Dewar, M.L. McKee, J. Comp. Chem. 4 (1983) 84. [137] L.P. Davis, et al., J. Comp. Chem. 2 (1981) 433. [138] M.J.S. Dewar, M.L. McKee, H.S. Rzepa, J. Am. Chem. Soc. 100 (1978) 3607. [139] M.J.S. Dewar, E.F. Healy, J. Comp. Chem. 4 (1983) 542. [140] M.J.S. Dewar, G.L. Grady, J.J.P. Stewart, J. Am. Chem. Soc. 106 (1984) 6771. [141] M.J.S. Dewar, et al., Organometallics 4 (1985) 1964. [142] M.J.S. Dewar, E.G. Zoebisch, E.F. Healy, J. Am. Chem. Soc. 107 (1985) 3902. [143] M.J.S. Dewar, C.H. Reynolds, J. Comp. Chem. 2 (1986) 140. [144] J.J.P. Stewart, J. Comput. Chem. 10 (1989) 209. [145] J.J.P. Stewart, J. Comput. Chem. 10 (1989) 221. [146] D. Cornell, P. Cieplak, C.I. Bayly, I.R. Gould, K.M. Merz Jr., D.M. Ferguson, D.C. Spellmeyer, T. Fox, J.W. Caldwell, P.A. Kollman, J. Am. Chem. Soc. 117 (1995) 5179. [147] S.L. Mayo, B.D. Olafson, W.A. Goddard, J. Phys. Chem. 94 (1990) 8897. [148] A.K. Rappe, C.J. Casewit, K.S. Colwell, W.A. Goddard III, W.M. Skiff, J. Am. Chem. Soc. 114 (1992) 10024. [149] A.K. Rappe, W.A. Goddard III, J. Phys. Chem. 95 (1991) 3358. [150] A. Lesar, S. Prebil, M. Hodoscek, J. Chem. Inform. Comput. Sci. 42 (4) (2002) 853. [151] G.A. Voth, Abstracts of Papers, 223rd ACS National Meeting, Orlando, FL, United States, April 7 –11 2002. [152] H.B. Schlegel, J.M. Millam, S.S. Iyengar, G.A. Voth, A.D. Daniels, G.E. Scuseria, M.J. Frisch, J. Chem. Phys. 114 (2001) 9758. [153] C.K. Sohn, E.K. Ma, C.K. Kim, H.W. Lee, I. Lee, New J. Chem. 25 (6) (2001) 859. [154] J. Saltiel, O. Dmitrenko, W. Reischl, R.D. Bach, J. Phys. Chem. A 105 (15) (2001) 3934. [155] X. Zhang, Y.H. Ding, Z.S. Li, X.R. Huang, C.C. Sun, Chem. Phys. Lett. 330 (5– 6) (2000) 577. [156] X. Zhang, Y.H. Ding, Z.S. Li, X.R. Huang, C.C. Sun, J. Phys. Chem. A 104 (36) (2000) 8375.

20


[157] Y. Yamamori, K. Takahashi, T. Inomata, J. Phys. Chem. A 103 (44) (1999) 8803. [158] J.R. Flores, C.M. Estevez, L. Carballeira, I.P. Juste, J. Phys. Chem. A 105 (19) (2001) 4716. [159] C.M. Estevez, O. Dmitrenko, J.E. Winter, R.D. Bach, J. Organic Chem. 65 (25) (2000) 8629. [160] B.K. Carpenter, J. Phys. Chem. A 105 (19) (2001) 4585. [161] K.K. Zhu, Z.T. Gan, K.H. Su, Y.B. Wang, Z.Y. Wen, X. Huaxue, 58(5) (2000) 515. [162] T. Yamada, M. Steiger, P.H. Taylor, J.W. Bozzelli, Chemical and Physical Processes in Combustion (1999) 312. [163] K. van Alem, G. Lodder, H. Zuilhof, J. Phys. Chem. A 104 (12) (2000) 2780. [164] J.F.K. Muller, R. Batra, J. Organometallic Chem. 584 (1) (1999) 27. [165] P.R. Briddon, R. Jones, Comput. Simulation Mater. Atomic Level (2000) 131–171. [166] Y. Liu, M.E. Tuckerman, Department of Chemistry, New York University, New York, NY, USA, Abstracts of Papers, 224th ACS National Meeting, Boston, MA, United States, 18– 22, 2002. [167] C.K. Siu, Z.F. Liu, J.S. Tse, J. Am. Chem. Soc. 124 (36) (2002) 10846. [168] C. Chen, M.H. Liu, T. Ta-His, H. Jishu, Taiwan 18 (1) (2002) 83– 91. [169] J.-C. Zhang, H.-Y.Wang, Y.-J. Tang, Z.-H. Zhu, W.-D. Wu, X. Wuli, 51(6) (2002) 1221–1226. [170] F. Turecek, J. Phys. Chem. A 102 (24) (1998) 4703. [171] M. Segovia, O.N. Ventura, Chem.Phys. Lett. 277 (5–6) (1997) 490. [172] B.S. Jursic, Int. J. Quant. Chem. 65 (1) (1997) 75. [173] J. Harmer, S. van Doorslaer, I. Gromov, M. Broering, G. Jeschke, A. Schweiger, J. Phys. Chem. B 106 (10) (2002) 2801. [174] S. Parthiban, G. de Oliveira, J.M.L. Martin, J. Phys. Chem. A 105 (5) (2001) 895. [175] B. Wang, H. Hou, Y. Gu, J. Chem. Phys. 112 (19) (2000) 8458. [176] A.F. Jalbout, Int. J. Quant. Chem. 86 (2002) 541. [177] T.P.W. Jungkamp, J.H. Seinfeld, J. Chem. Phys 107 (5) (1997) 1513. [178] B.S. Jursic, Theochem 427 (1998) 157. [179] B.F. Yates, Theochem 506 (2000) 223. [180] B. Menendez, V.M. Rayon, J.A. Sordo, A. Cimas, C. Barrientos, A. Largo, J. Phys. Chem. A 105 (43) (2001) 9917. [181] S. Ackloo, J.K. Terlouw, P.J.A. Ruttink, P.C. Burgers, Rapid Commun. Mass Spectrometry 15 (14) (2001) 1152. [182] D. Kovacs, J.E. Jackson, J. Phys. Chem. A 105 (32) (2001) 7579. [183] J.F. Gal, M. Decouzon, P.C. Maria, A.I. Gonzalez, O. Mo, M. Yanez, S. el Chaouch, J.C. Guillemin, J. Am. Chem. Soc. 123 (26) (2001) 6353. [184] S. Han, M.C. Hare, S.R. Kass, Int. J. Mass Spectrometry 201 (1–3) (2000) 101. [185] P. Burk, I.A. Koppel, A. Rummel, T. Aleksander, J. Phys. Chem. A 104 (7) (2000) 1602. [186] I.P.R. Moreira, Theochem 466 (1999) 119. [187] J.W. Bozzelli, C. Sheng, J. Phys. Chemi. A 106 (7) (2002) 1113. [188] A.P. Scott, M.S. Platz, L. Radom, J. Am. Chem. Soc. 123 (25) (2001) 6069. [189] T.V. Duncan, C.E. Miller, J. Chem. Phys. 113 (13) (2000) 5138. [190] C.E. Miller, J.S. Francisco, J. Phys. Chem. A 105 (4) (2001) 750. [191] X. Zhou, J. Li, X. Zhao, Y. Tian, L. Zhang, Y. Chen, C. Chen, S. Yu, X. Ma, Phys. Chem. Chem. Phys. 3 (17) (2001) 3662. [192] H. Eyring, J. Chem. Phys 3 (1934) 107. [193] J. Fabian, A. Krebs, D. Schoenemann, W. Schaefer, J. Org. Chem. 65 (26) (2000) 8940. [194] C. Sheng, J.W. Bozzelli, Int. J. Chem. Kinetics 32 (10) (2000) 623. [195] W.C. Kreye, P.G. Seybold, Chem. Phys. Lett. 335 (3–4) (2001) 257. [196] J.W. Bozzelli, C.J. Chen, C. Sheng, A.M. Dean, Preprints Symp.— Am. Chem. Soc. Div. Fuel Chem. 47 (1) (2002) 219. [197] W.C.N. Hon, Z.D. Chen, Z.F. Liu, J. Phys. Chem. A 106 (29) (2002) 6792– 6801. [198] P. Devolder, C. Fittschen, L. ElMaimouni, H. Hippler, B. Viscolcz, Laboratoire de Cinetique et Chimie de la Combustion, CNRS UMR

[199] [200] [201] [202] [203] [204]

[205] [206] [207] [208] [209] [210] [211] [212] [213] [214] [215] [216]

[217] [218]

[219] [220] [221] [222] [223] [224] [225] [226] [227] [228] [229] [230] [231]

[232] [233] [234]

8522, Universite de Lille, Villeneuve d’Ascq, in: P. Midgley, M. Reuther, M. Williams (Eds.), Transport and Chemical Transformation in the Troposphere, Proceedings of EUROTRAC Symposium, 6th, Garmisch – Partenkirchen, Germany, March 27 – 31, 2000 (2001), Meeting Date, 2000, pp. 397 –400. G.H. Yang, H.X. Zhang, A.Q. Tang, X. Huaxue, 59(4) (2001) 486. C.J. Chen, J.W. Bozzelli, J. Phys.Chem. A 104 (21) (2000) 4997. A.F. Jalbout, A.M. Darwish, H.Y. Alkahby, Theochem 585 (2002) 199. B. Militzer, D.M. Ceperty, Phys. Rev. Lett. 85 (2000) 1890. P. Politzer, P. Lane, M.C. Concha, 691 (JANNAF 36th Combustion Subcommittee Meeting), vol. 2, 1999, pp. 331. G.A. Petersson, D.K. Malick, J.A. Keith, Y.T. Chen, J.W. Ochterski, J.A. Montgomery, Jr., M.J. Frisch, Hall-Atwater Laboratories of Chemistry, Wesleyan University, Middletown, CT, USA, Book of Abstracts, 218th ACS National Meeting, New Orleans, August 22 –26, 1999, PHYS-412. C. Majumder, V. Kumar, H. Mizuseki, Y. Kawazoe, Chem. Phys. Lett. 356 (1–2) (2002) 36. A.F. Jalbout, S. Fernandez, Theochem 584 (2002) 169. A.F. Jalbout, S. Fernandez, H. Chen, Theochem 584 (2002) 43. W. Koch, M.C. Holthausen, A Chemist’s Guide to Density Functional Theory, second ed., Wiley-VCH, Weinheim, 2002. P.M. Mayer, Chem. Phys. Lett. 314 (3–4) (1999) 311. B.S. Jursic, Theochem 428 (1998) 175. H. Valdes, J.A. Sordo, J. Comput. Chem. 23 (4) (2002) 444. A. Damin, S. Bordiga, A. Zecchina, C. Lamberti, J. Chem. Phys. 117 (1) (2002) 226. A. Luna, M. Alcami, O. Mo, M. Yanez, de M. Tortajada, Int. J. Mass Spectrometry 217 (1–3) (2002) 119. S.I. Solling, S.B. Wild, L. Radom, Inorg. Chem. 38 (26) (1999) 6049. A. Bagno, F. Terrier, J. Phys. Chem. A 105 (26) (2001) 6537. L. Wang, J. Zhang, Book of Abstracts, 219th ACS National Meeting, San Francisco, CA, March 26–30, 2000, PHYS-243, American Chemical Society, Washington, DC, 2000. M. Mascal, A. Armstrong, M.D. Bartberger, J. Am. Chem. Soc. 124 (22) (2002) 6274. B.S. Jursic, Book of Abstracts, 214th ACS National Meeting, Las Vegas, NV, September 7– 11 1997, American Chemical Society, Washington, DC, 1997. X.Y. Chen, C.X. Zhao, Y. Ping, G.Z. Ju, Int. J. Quant. Chem. 85 (3) (2001) 127. Z. Latajka, Y. Bouteiller, J. Chem. Phys. 101 (1994) 3793. C. Tuma, A.D. Boese, N.C. Handy, Phys. Chem. Chem. Phys. 1 (1999) 3939. R.A. Klein, J. Comp. Chem. 23 (6) (2002) 585. P. Sanz, M. Yanez, O. Mo, J. Phys. Chem. A 106 (18) (2002) 4661. P.R. Rablen, J.W. Lockman, W.L. Jorgensen, J. Phys. Chem. A 102 (21) (1998) 3782. D.M. Birney, J. Am. Chem. Soc. 122 (44) (2000) 10917. S. Arulmozhiraja, P. Kolandaivel, O. Ohashi, J. Phys. Chem. A 103 (16) (1999) 3073. T.P.M. Goumans, A.W. Ehlers, M.J.M. Vlaar, S.J. Strand, K. Lammertsma, J. Organometallic Chem. 643 –644 (2002) 369. J.E. Walker, P.A. Adamson, S.R. Davis, Theochem 487 (1–2) (1999) 145. B.G. Johnson, M.J. Frisch, Chem. Phys. Lett. 216 (1993) 133. R.E. Stratmann, J.C. Burant, G.E. Scuseria, M.J. Frisch, J. Chem. Phys. 106 (1997) 10175. A.F. Jalbout, Abstracts of Papers, 222nd ACS National Meeting, Chicago, IL, United States, August 26 – 30, 2001, PHYS-305, American Chemical Society, Washington, DC, 2001. M. Remko, J. Sefcikova, Theochem 528 (2000) 287. V.V. Sliznev, S.B. Lapshina, G.V. Girichev, J. Struct. Chem. 43 (1) (2002) 47 –55. Translation of Zhurnal Strukturnoi Khimii. A. Lesar, M. Hodoscek, J. Chem. Inform. Comput. Sci. 42 (3) (2002) 706.

A.F. Jalbout et al. / Journal of Molecular Structure (Theochem) 671 (2004) 1–21 [235] J.C. Sancho-Garcia, A.J. Perez-Jimenez, J.M. Perez-Jorda, F. Moscardo, Mol. Phys. 99 (1) (2001) 47. [236] P. Redondo, J.R. Redondo, A. Largo, Theochem 505 (2000) 221. [237] H. Somnitz, R. Zellner, Phys. Chem. Chem. Phys. 3 (12) (2001) 2352. [238] A.F. Jalbout, Chem. Phys. Lett. 340 (5–6) (2001) 571. [239] H. Somnitz, R. Zellner, Phys. Chem. Chem. Phys. 3 (12) (2001) 2352. [240] B. Wang, H. Hou, Y. Gu, J. Phys. Chem. A 105 (1) (2001) 156. [241] J.W. Bozzelli, D. Jung, J. Phys. Chem. A 105 (16) (2001) 3941. [242] H. Somnitz, R. Zellner, Phys. Chem. Chem. Phys. 2 (9) (2000) 1899. [243] K.K. Baeck, K. Kyoung, H. Choi, S. Iwata, J. Phys. Chem. A 103 (34) (1999) 6772. [244] Q. Li, K. Wu, H.S. Lu, S.D. Xuebao, Z. Kexueban, 25 (2) 2002 189–190. [245] J.C. Traeger, Int. J. Mass Spectrometry 210/211 (1–3) (2001) 181. [246] J.C. Traeger, M. Djordjevic, Eur. Mass Spectrometry 5 (5) (1999) 319. [247] J.L.M. Abboud, O. Castano, J.Z. Davalos, R. Gomperts, Chem. Phys.Lett. 337 (4–6) (2001) 327. [248] L.A. Curtiss, P.C. Redfern, D.J. Frurip, Rev. Comput. Chem. 15 (2000) 147. [249] K.B. Wiberg, J. Ochterski, A. Streitwieser, J. Am. Chem. Soc. 118 (35) (1996) 8291. [250] J.C. Rienstra-Kiracofe, G.S. Tschumper, H.F. Schaefer III, S. Nandi, G.B. Ellison, Chem. Rev. (Washington, DC) 102 (1) (2002) 231. [251] A. Larry, P. Curtiss, G. Redfern, K. Kaghavachari, J.A. Pople, J. Chem. Phys. 109 (1998) 42. [252] A.G. Baboul, L.A. Curtiss, P.C. Redfern, K. Raghavachari, J. Chem. Phys. 110 (16) (1999) 7650. [253] J.C. Rienstra-Kiracofe, G.S. Tschumper, H.F. Schaefer III, S. Nandi, G.B. Ellison, Chem. Rev. (Washington DC) 102 (1) (2002) 231. [254] H.R. Hu, A. Tian, N.B. Wong, W.K. Li, J. Phys. Chem. A 105 (45) (2001) 10372. [255] C.F. Tormena, N.S. Amadeu, R. Rittner, R.J. Abraham, J. Chem. Soc. Perkin Trans. II 4 (2002) 773. [256] M. Remko, O.A. Walsh, W.G. Richards, Chem. Phys. Lett. 336 (1–2) (2001) 156. [257] J.M.M. Vlaar, A.W. Ehlers, F.J.J. De Kanter, M. Schakel, L.A. Spek, M. Lutz, N. Sigal, Y. Apeloig, K. Lammertsma, Angewandte Chem., Int. Ed. 39 (22) (2000) 4127. [258] B. Smith, Phys. Chem. Chem. Phys. 2 (23) (2000) 5383. [259] M.R. Cameron, G.B. Bacskay, J. Phys. Chem. A 104 (47) (2000) 11212. [260] R. Notario, O. Castano, J.L.M. Abboud, R. Gomperts, L.M. Frutos, R. Palmeiro, J. Org. Chem. 64 (25) (1999) 9011. [261] D.M. Koch, N.H. Khieu, G.H. Peslherbe, J. Phys. Chem. A 105 (14) (2001) 3598. [262] C.J. Cobos, C. Theochem 581 (2002) 17. [263] R. Sumathi, H.H. Carstensen, W.H. Green Jr., J. Phys. Chem. A 105 (28) (2001) 6910. [264] D. Jung, J.W. Bozzelli, J. Phys. Chem. A 105 (22) (2001) 5420–5430. [265] D. Jung, C.J. Chen, J.W. Bozzelli, J. Phys. Chem. A 104 (42) (2000) 9581. [266] L. Wang, J. Zhang, Theochem 581 (2002) 129. [267] P. Politzer, P. Lane, M.C. Concha, J. Phys. Chem., A 103 (10) (1999) 1419–1425. [268] H. Wang, K. Brezinsky, J. Phys. Chem. A 102 (9) (1998) 530. [269] C.W. Bauschlicher Jr., H. Partridge, Chem.Phys. Lett. 240 (5–6) (1995) 533.

21

[270] T.L. Nguyen, G.S. Kim, A.M. Mebel, M.N. Minh, Chem. Phys. Lett. 349 (5– 6) (2001) 571. [271] S. Parthiban, J.M.L. Martin, J.F. Liebman, Preprint Arch. Phys. (2001) 1. [272] R.M.L. Savedra, J.C. Pinheiro, O. Treu Filho, R.T. Kondo, Theochem 587 (2002) 9. [273] L. Zhu, J.W. Bozzelli, J. Phys. Chem. A 106 (2) (2002) 345. [274] K.R. Shamasundar, E. Arunan, J. Phys. Chem. A 105 (37) (2001) 8533. [275] S. Arulmozhiraja, S. Tadatake, A. Yabe, J. Comput. Chem. 22 (9) (2001) 923. [276] C.-J. Chen, J.W. Bozzelli, J. Phys. Chem. A 104 (43) (2000) 9715–9732. [277] L. Zhu, C.J. Chen, J.W. Bozzelli, J. Phys. Chem. A 104 (40) (2000) 9197. [278] C. Chen, Int. J. Quantum Chem. 80 (1) (2000) 27. [279] C.J. Chen, J.W. Bozzelli, Chem. Phys. Process Combust. (1999) 37. [280] C.J. Chen, J.W. Bozzelli, J. Phys. Chem. A 103 (48) (1999) 9731. [281] P.M. Mayer, M.S. Taylor, M.W. Wong, L. Radom, J. Phys. Chem. A 102 (35) (1998) 7074. [282] C. Trindle, K. Romberg, J. Phys. Chem. A 102 (1) (1998) 270. [283] H.L. Schmider, A.D. Becke, J. Chem. Phys. 108 (1998) 9624. [284] Curtiss, et al., J. Chem. Phys. 106 (1997) 1063. [285] H.L. Schmider, A.D. Becke, J. Chem. Phys. 109 (19) (1998) 8188. [286] A.D. Becke, J. Chem. Phys. 107 (1997) 8554. [287] L.A. Curtiss, et al., J. Chem. Phys. 106 (1997) 1063. [288] H.L. Schmider, A.D. Becke, J. Chem. Phys. 108 (23) (1998) 9624. [289] P.C. Redfern, J.P. Blaudeau, L.A. Curtiss, J. Phys. Chem. A 101 (46) (1997) 8701. [290] B.J. Smith, L. Radom, Chem. Phys. Lett. 245 (1) (1995) 123. [291] B.J. Smith, L. Radom, Chem. Phys. Lett. 231 (4–6) (1994) 345. [292] J.Y. Choi, C.K. Kim, C.K. Kim, I. Lee, J. Phys. Chem. A 106 (23) (2002) 5709. [293] D.J. Henry, C.J. Parkinson, P.M. Mayer, L. Radom, J. Phys. Chem. A 105 (27) (2001) 6750. [294] C.F. Wilcox, Y.X. Zhang, S.H. Bauer, Theochem 538 (2001) 67. [295] N.B. Wong, Y.S. Cheung, D.T. Wu, Y. Ren, A. Tian, W.K. Li, J. Phys. Chem. A 104 (25) (2000) 6077. [296] P.C. Redfern, P. Zapol, L.A. Curtiss, K. Raghavachari, J. Phys. Chem. A 104 (24) (2000) 5850. [297] L.A. Curtiss, K. Raghavachari, P.C. Redfern, J.A. Pople, J. Chem. Phys. 112 (17) (2000) 7374. [298] J.P. Senosiain, J.H. Han, C.B. Musgrave, D.M. Golden, Faraday Discuss. 119 (2001) 173 Combustion chemistry: elementary reactions to macroscopic processes. [299] S. Sekusak, G.F. Frenking, Theochem 541 (2001) 17. [300] B.S. Jursic, Theochem 487 (1–2) (1999) 193. [301] G.A. Petersson, D.K. Malick, J.A. Keith, Y.T. Chen, J.W. Ochterski, J.A. Montgomery, Jr., M.J., Frisch, Book of Abstracts, 218th ACS National Meeting, New Orleans, August 22–26, 1999. [302] K. van Alem, E.J.R. Sudhoelter, H. Zuilhof, J. Phys. Chem. A 102 (52) (1998) 10860. [303] G.A. Petersson, D.K. Malick, W.G. Wilson, J.W. Ochterski, J.A. Montgomery Jr., M.J. Frisch, J. Chem. Phys. 109 (24) (1998) 10570. [304] J.A. Montgomery Jr., J.W. Ochterski, M.J. Frisch, G.A. Petersson, Lorentzian, C.T. Haven, Book of Abstracts, 216th ACS National Meeting, Boston, August 23–27 (1998), PHYS-353, American Chemical Society, Washington, DC, 1998. [305] A.F. Jalbout and co-workers, work in progress.

Gaussian-based computations in molecular science

Gaussian-based computations in molecular science

Suggest Documents

Gaussian-based computations in molecular science

Matrix Computations - Computer Science

Symbolic Computations in Science and Engineering - wseas

Molecular complexes theoretical computations between methanol and ...

Molecular computations for reactions and phase transitions

Computations in Commutative Algebra

An Appropriate Algorithm in Parallel Computations for ... - Science Direct

Recursive Computations in Spreadsheets

Petascale computations for Large-scale Atomic and Molecular collisions

Computations in Propositional Logic

Understanding Rulelog Computations in Silk

COMPREHENSIVE MOLECULAR INSECT SCIENCE

SINGULAR: LETTERPLACE, COMPUTATIONS IN NON ...

Molecular Imprinting Applications in Forensic Science - DiVA

Impact of currents on surface flux computations and ... - Ocean Science

Biomedical Computations on the Grid - Computational Science @ UvA

Numerical computations of resistance of high speed ... - Science Direct

Interactive Visualization of Earth and Space Science Computations

Frequency and time domain flutter computations of a ... - Science Direct

Time-Optimal Proximity Graph Computations on ... - Science Direct

Parallel and distributed computations of maximum ... - Science Direct

Analytical computations of dynamic behaviour of pin ... - Science Direct

Survey Computations Textbook

Matrix Computations – COMP6839 - UPRM