Density functional theory optimized basis sets for ...

THE JOURNAL OF CHEMICAL PHYSICS 126, 044108 共2007兲

Density functional theory optimized basis sets for gradient corrected functionals: 3d transition metal systems Patrizia Calaminici,a兲 Florian Janetzko, Andreas M. Köster, Roberto Mejia-Olvera, and Bernardo Zuniga-Gutierrez Departamento de Química, CINVESTAV, Avenida Instituto Politécnico Nacional 2508, Apartado postal 14-740, México, Distrito Federal 07000, Mexico

共Received 24 October 2006; accepted 12 December 2006; published online 30 January 2007兲 Density functional theory optimized basis sets for gradient corrected functionals for 3d transition metal atoms are presented. Double zeta valence polarization and triple zeta valence polarization basis sets are optimized with the PW86 functional. The performance of the newly optimized basis sets is tested in atomic and molecular calculations. Excitation energies of 3d transition metal atoms, as well as electronic configurations, structural parameters, dissociation energies, and harmonic vibrational frequencies of a large number of molecules containing 3d transition metal elements, are presented. The obtained results are compared with available experimental data as well as with other theoretical data from the literature. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2431643兴 I. INTRODUCTION

Transition metal 共TM兲 systems have received considerable attention in the last decades due to their potential application for the design of novel materials and in catalytic processes.1–8 Although the theoretical study of these systems has been simplified by the development of molecular density functional theory 共DFT兲,9–15 many problems still remain unsolved. For example, the accurate calculation of ground and excited state energies for 3d transition metals has proven to be a very difficult problem for electronic structure theory.16,17 In particular, density functional methods tend to overestimate the stability of the s1dn states compared to the s2dn−1 ones.18 This problem has serious consequences for the reliable calculation of binding energies of transition metal clusters. Moreover, the computed atomic hybridization energies are often incorrect, which may lead to wrong ground state structures 共see Ref. 17 and references therein兲. For this reason DFT calculations should be combined with experimental data for the reliable structure determination of free and substituted TM clusters.17,19–24 In particular, pulsed-field ionization zero electron kinetic energy 共PFIZEKE兲 photoelectron spectroscopy as well as vibrational resolved negative ion photoelectron spectroscopy provide sufficient information in combination with DFT calculations to unequivocally assign the structure of free and substituted TM clusters. A good example represents the structure determination of V3 from a PFI-ZEKE spectrum with well resolved vibrational bands25 and corresponding DFT calculations.17 A rather interesting detail of this work was the performance of the double zeta valence polarization 共DZVP兲 basis set of vanadium which was optimized for gradient corrected functionals. It turned out that such a basis set optimization is mandatory for the correct ground state prediction of the vaa兲

Author to whom correspondence should be addressed. Electronic mail: [email protected]

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nadium trimer. Only with the vanadium DZVP basis set optimized for the generalized gradient approximation 共GGA兲 from Perdew and Wang26,27 共PW86兲 the atomic ground and low-lying excited states of the vanadium atom can be correctly reproduced in the calculation. Moreover, the calculated s – d energy splitting is in very good agreement with the experimental data.17 Based on this result we developed GGA optimized basis sets for all 3d TM elements.28 Even so the basis sets are developed with the PW86 GGA functional, they are transferable to other gradient corrected functionals as shown in Ref. 28. Very recently it was shown that these basis sets perform well for single charged first row TM cations,29 too. In this article we critically evaluate the performance of GGA optimized basis sets in the calculation of 3d transition metal atoms and molecules. For this purpose we present double 共DZVP兲 and newly developed triple zeta valence polarization 共TZVP兲 basis sets for 3d elements optimized for the PW86 GGA functional. We show that these basis sets improve systematically the calculated s2dn−1 → s1dn atomic excitation energies compared with their counterparts optimized for local functionals. We then analyze the performance of these basis sets in the calculation of homonuclear 3d TM dimers. Here we compare electronic ground states, bond lengths, frequencies, and dissociation energies with experimental gas phase data as well as other theoretical results. We conclude our analysis of GGA optimized basis sets with the comparison of theoretical and experimental structure data of a test set of 78 TM compounds 共TM1 set兲 for which experimental gas phase structure data are available. The article is organized as follows. In the next two sections the details of the basis set optimization and electronic structure calculations are given. In Sec. III the 3d transition metal atom calculations are presented. Section IV discusses the homonuclear 3d TM dimer calculations. In Sec. V the

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performance of the GGA optimized basis sets on the TM1 test set is analyzed. Final conclusions are drawn in the last section.

II. BASIS SET OPTIMIZATION

The GGA optimization of DZVP and TZVP basis sets was performed with the linear combination of Gaussian-type orbitals Kohn-Sham code deMon2k.30 The newly developed basis sets, from now on named DZVP-GGA and TZVPGGA, were obtained by recontraction of the orbital basis sets originally developed for the local density approximation 共LDA兲.31 From now on we call these basis sets DZVP-LDA and TZVP-LDA, respectively. The DZVP and TZVP LDA and GGA basis sets are available for all 3d elements. The exponents and contraction coefficients of the DZVPGGA and the newly developed TZVP-GGA basis sets for 3d elements are given in the supplementary material of this article.32 These basis sets are also available in the basis set file of the deMon2k code and can be downloaded at http://www.demon-software.com/public_html/download .html#basissets. Following our previous works,17,28 the PW86 共Refs. 26 and 27兲 functional was used for the optimization of the contraction coefficients. In order to avoid the calculation of fourcenter electron repulsion integrals, the variational fitting of the Coulomb potential33 was employed. The used GEN-A2 Hermite auxiliary functions were automatically generated for the specified basis set. A detailed description of the automatic generation of auxiliary functions in deMon2k is given in the Appendix of this article. The contraction coefficients were optimized by atomic ground state calculations with the fully decontracted basis sets. The ground state configurations were taken from Ref. 34. In these calculations spherical atomic orbitals are used. In order to avoid spin contamination, the restricted openshell Kohn-Sham 共ROKS兲 method was employed. The atomic states were represented by the following one determinant open-shell configurations: d1共 2D兲: 共dz2兲1 , d2共 3F兲: 共dz2兲1共dx2−y2兲1 , d3共 4F兲: 共dxy兲1共dxz兲1共dyz兲1 ,

d9共 2D兲: 共dz2兲1共dx2−y2兲2共dxy兲2共dxz兲2共dyz兲2 . For the d1, d4, d6, and d9 configurations the orbital occupancies are purely arbitrary, but for the d2, d3, d7, and d8 configurations the orbital occupancies must be as given above to yield the correct spatial symmetry.35 The employed PW86 functional was numerically integrated on an adaptive grid.36,37 For the optimization of the contraction coefficients, the rotation of the Lebedev grids was disabled in order to avoid symmetry breaking due to the mixing of s and d atomic orbitals. In the atomic calculations the grid accuracy was set to 10−7. It should be noted that for nonsymmetric occupied d configurations large angular grids are necessary. The settings described here for the atomic calculations guarantee self-consistent field 共SCF兲 convergences of 10−8 or better with correct spatial symmetry. III. COMPUTATIONAL DETAILS

All molecular structures were optimized with the local exchange-correlation functional of Vosko-Wilk-Nusair38 共VWN兲, as well as with the gradient corrected PW86 共Refs. 26 and 27兲 functional and with the exchange-correlation contributions proposed by Becke and Lee, Yang and Parr39,40 共BLYP兲. The exchange-correlation potential for the structure optimization was calculated with the orbital density or the auxiliary function density. We name these two approaches BASIS and AUXIS, respectively. The AUXIS approach was recently implemented in the deMon2k code41 and has proven reliable for structure optimizations and energy calculations. Because the auxiliary function density is a linear combination of primitive Hermite Gaussian functions, the density calculation and, thus, the exchange-correlation potential calculation at each grid point scale linear. This is particularly useful in large scale calculations with 10 000 or more basis functions.42 For the structure optimization a quasi-Newton method in internal redundant coordinates with analytic energy gradients was used.43 The convergence was based on the Cartesian gradient and displacement vectors with a threshold of 10−4 and 10−3 a.u., respectively. A vibrational analysis was performed at the optimized geometries. The second derivatives were calculated by numerical differentiation 共two-point finite difference兲 of the analytic energy gradients using a displacement of 0.001 a.u. from the optimized geometry for all 3N coordinates. The harmonic frequencies were obtained by diagonalizing the mass-weighted Cartesian force constant matrix.

d4共 5D兲: 共dx2−y2兲1共dxy兲1共dxz兲1共dyz兲1 , IV. ATOMS

d 共 S兲: 共dz2兲共dx2−y2兲共dxy兲共dxz兲共dyz兲 , 5 6

1

1

1

1

1

d6共 5D兲: 共dz2兲2共dx2−y2兲1共dxy兲1共dxz兲1共dyz兲1 , d7共 4F兲: 共dz2兲2共dx2−y2兲2共dxy兲1共dxz兲1共dyz兲1 , d8共 3F兲: 共dz2兲1共dx2−y2兲1共dxy兲2共dxz兲2共dyz兲2 ,

The calculated s2dn−1 → s1dn excitation energies of 3d transition metals are listed in Table I and compared graphically in Fig. 1 with the corresponding experimental values.34 The excitation energies were calculated with the DZVP-LDA basis sets optimized for local functionals31 and the DZVPGGA basis sets optimized for gradient corrected functionals. In both cases, the PW86 functional26,27 was used. Table I shows that the DZVP-LDA basis set predicts for several 3d


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DFT optimized basis sets

TABLE I. Calculated and experimental s2dn−1 → s1dn excitation energies 共in eV兲 for 3d transition metals. The used configurations for the s2dn−1 and s1dn states are given for each atom, too. The calculated values were obtained employing the PW86 functional. The experimental values are given for comparison, too. s2dn−1

Atom Sc Ti V Cr Mn Fe Co Ni Cu

2

D F 4 F 5 D 6 S 5 D 4 F 3 F 2 D 3

3d14s2 3d24s2 3d34s2 3d44s2 3d54s2 3d64s2 3d74s2 3d84s2 3d94s2

s 1d n 4

F F 6 D 7 S 6 D 5 F 4 F 3 D 2 S 5

3d24s1 3d34s1 3d44s1 3d54s1 3d64s1 3d74s1 3d84s1 3d94s1 3d104s1

DZVP-LDA

DZVP-GGA

Expt.a

0.37 −0.14 −0.47 −1.80 0.49 −0.74 −0.68 −0.59 −2.04

0.77 0.47 0.22 −1.74 1.34 0.05 0.25 −0.39 −1.98

1.43 0.80 0.25 −0.99 2.16 0.87 0.41 −0.03 −1.51

a

Reference 34.

elements the wrong ground state. Compared to the DZVPLDA basis set the new DZVP-GGA basis set improves systematically the accuracy of the low-lying energy levels of the 3d TM atoms. As a result all 3d element ground states are correctly predicted with the DZVP-GGA basis set in combination with the PW86 functional. The general tendency of the DZVP-LDA basis set to favor s1dn over s2dn−1 states is eliminated by the GGA optimized DZVP basis set. Figure 1 shows that the general sawtooth behavior in the excitation energy is faithfully reproduced by the ROKS DZVP-GGA/PW86 method for the 3d TM atom series. This method gives excellent agreement with the experimental data for V and Co 共see Fig. 1兲. Deviations between 0.3 and 0.8 eV are observed for the other 3d TM atoms. Some of these deviations can be reduced with the TZVP-GGA basis set optimized for gradient corrected functionals. However, the relative large discrepancies for Cr, Mn, and Fe remain also with larger GGA optimized basis sets. This indicates a principle problem in the functional rather than in the basis set. V. DIMERS

The electronic states, bond lengths, harmonic frequencies, and dissociation energies of the homonuclear 3d TM dimers are presented and discussed in this section. For this analysis we employed the DZVP-LDA/VWN, DZVP-LDA/ BLYP, DZVP-LDA/PW86, DZVP-GGA/VWN, DZVP-

FIG. 1. Calculated and experimental s2dn−1 → s1dn excitation energies for the 3d transition metal series.

GGA/BLYP, DZVP-GGA/PW86, TZVP-GGA/VWN, TZVP-GGA/BLYP, and TZVP-GGA/PW86 methods.

A. Electronic states

In this section we present the spectroscopic states and the corresponding electronic configurations of the studied 3d dimer ground states obtained with the ROKS DZVP-GGA/ PW86 method. For all dimers several states and multiplicities have been studied in order to find the 共theoretically predicted兲 ground state structure. We compare our results with available experimental data and previous theoretical studies. The first diatomic homonuclear molecule of the 3d block, the scandium dimer, corresponds experimentally to a 5 − ⌺u state.2 In agreement with the experimental data we find a 5 ⌺u state with the electronic configuration 共4s␴g兲2共3d␲u兲2共3d␴g兲1共4s␴*u兲1 as the Sc2 ground state. Our theoretical result agrees well with other theoretical studies.44–47 The determination of the titanium dimer ground state is very difficult because of many low-lying states. We predict the ground state of the titanium dimer as a 3⌬g state with a 共4s␴g兲2共3d␴g兲1共3d␲u兲4共3d␦g兲1 electronic configuration. The same result was found in other theoretical studies,45–47 too. The ground state of the vanadium dimer was experimentally assigned as a 3⌺−g state.2 In previous theoretical works the electronic configuration of this state was predicted as 共3d␲u兲4共3d␴g兲2共3d␦g兲2共4s␴g兲2.44,45 Our result is in perfect agreement with these predictions. The experimental ground state of the chromium dimer corresponds to a 1⌺+g state.2 Our study predicts a 1⌺g ground state with a 共3d␴g兲2共3d␲u兲4共3d␦g兲4共4s␴g兲2 electronic configuration, in agreement with previous theoretical studies.44–47 For the manganese dimer a 1⌺+g ground state is experimentally assigned.2 This is a peculiar van der Waals molecule and, therefore, complicated to describe with current DFT approaches. In the literature a controversy exists if the manganese dimer ground state is either a 1⌺g, 11⌸u, or 11⌺g state.44 We find a 11⌺g ground state with the electronic configuration 共4s␴g兲2共3d␴g兲2共3d␲u兲2共3d␦g兲2共3d␦*u兲2共3d␴*u兲1 * 2 * 1 共3d␲g兲共4s␴u兲 . Only slightly higher in energy lies the 11⌸u state with the electronic configuration


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TABLE II. Calculated bond lengths 共in Å兲 of 3d transition metal dimers. The available experimental values are reported for comparison, too. Sc2 ⌺u

5

Ti2 ⌬g

3

V2 ⌺g

3

Cr2 ⌺g

1

Mn2 11 ⌺g

Fe2 ⌬u

Co2 5 ⌬g

7

Ni2 ⌺g

Cu2 1 ⌺g

3

DZVP-LDA

VWN BLYP PW86

2.526 2.630 2.633

1.916 1.989 1.992

1.776 1.816 1.816

1.642 1.678 1.680

2.643 2.760 2.756

1.961 2.020 2.012

1.953 2.243a 2.223a

2.074 2.168 2.151

2.208 2.304 2.280

DZVP-GGA

VWN BLYP PW86

2.619 2.685 2.669

1.900 1.976 1.980

1.759 1.803 1.802

1.641 1.678 1.680

2.663 2.769 2.736

1.986 2.044 2.033

1.983 2.275a 2.231a

2.107 2.194 2.177

2.212 2.307 2.282

TZVP-GGA

VWN BLYP PW86

2.594 2.674 2.658

1.886 1.968 1.973

1.751 1.796 1.797

1.624 1.663 1.666

2.614 2.710 2.681

1.956 2.017 2.008

1.960 2.218a 2.195a

2.056 2.133 2.119

2.169 2.263 2.240

¯

1.943b

1.770c

1.679d

2.020f

¯

2.154g

2.219h

Expt.

艋3.4e

a

Symmetry broken solution. Reference 52. c Reference 53. d Reference 54.

e

b

f

共4s␴g兲2共3d␲u兲3共3d␦g兲2共3d␴g兲1共3d␦*u兲2共3d␲*g兲2共3d␴*u兲2. Because the energy difference between these two states is less than 1 kJ/ mol, they are de facto degenerated. In fact, the bond length and frequency of the 11⌺g and 11⌸u manganese dimer are almost identical. The 11⌸u state has been assigned in several other DFT calculations as the manganese dimer ground state.44,46,48–50 Different theoretical studies assigned a 7⌬u ground state to the iron dimer.44–47 Our study confirms the 7⌬u ground state assignment. The corresponding electronic configuration is 共3d␴g兲2共3d␲u兲4共3d␦g兲3共3d␦*u兲2共4s␴g兲2共3d␲*g兲2共3d␴*u兲1, in agreement with previous theoretical studies.44–46 The cobalt dimer, Co2, is the homonuclear 3d dimer which is less understood, since experimental and theoretical information about this system are very limited. Previous theoretical calculations suggest either a 5⌬g or a 5⌺g ground state.45 We obtain a symmetry broken C⬁v 5⌬ ground state for the cobalt dimer with the ROKS DZVP-GGA/PW86 method. The frequency and binding energy of our optimized Co2 compare favorably with the available experimental results. The electronic configuration of this state is 共4s␴兲2共3d␴兲2共3d␲兲4共3d␦兲4共3d␦*兲3共3d␴*兲1共3d␲*兲2. The D⬁h 5 ⌺g and 5⌬g lie 4 and 6 kcal/ mol above this ground state, respectively. The ground state structure determination of the nickel dimer has been a matter of controversial debates. The experimental ground state structure is still unknown. Several theoretical investigations have been carried out in the last decade with the aim to resolve this controversy. In one of the most recent theoretical works, Yanagisawa et al.45 assign two triplet candidates, namely, 3⌺+u and 3⌺−g , as possible Ni2 ground states. In this work we find a 3⌺g ground state for Ni2 with the electronic configuration 共3d␴g兲2共3d␲u兲4共4s␴g兲2共3d␦g兲4 共3d␴*u兲2共3d␦*u兲4共3d␲*g兲2. The ground state of the copper dimer is experimentally well established as 1⌺+g .2 Cu2 is a unique 3d transition metal dimer in which the chemical bond is mainly formed by the 4s atomic orbitals.2,51 We predict a 1⌺g ground state for Cu2

with the electronic configuration 共3d␴g兲2共3d␲u兲4共3d␦g兲4 共3d␦*u兲4共3d␲*g兲4共3d␴*u兲2共4s␴g兲2. This is in agreement with the experimental observation.

Reference 55. Reference 56. g Reference 57. h Reference 58.

B. Bond lengths

The optimized geometries for the 3d TM dimers are reported in Table II. The available experimental values52–58 are given for comparison, too. Although there are no experimental values for the bond length of Sc2, an empirical estimation by Weisshaar derived from Badger’s rule gives a bond length of 2.29 Å.59 Unfortunately, as discussed by Weisshaar, Badger’s rule does not conform well for TM elements and it suffers from errors as large as 0.35 Å. Different theoretical methods have been employed in order to estimate the bond length of Sc2,46,60–65 too. Using MC-SCF/DZ, Hada et al. estimated a bond length of 2.50 Å.60 Later on Pettersson and co-workers, using CASSCF with contracted averaged coupled-pair functional and an atomic natural orbital 共ANO兲 basis set, obtained bond distances for Sc2 between 2.5 and 2.8 Å.61,62 Recently, Barden et al. performed a systematic DFT study on homonuclear 3d TM dimers and found a bond distance of around 2.6 Å for Sc2 using different GGA functionals.46 Previous DFT studies using BP86 共Refs. 63 and 64兲 and LDA 共Ref. 65兲 functionals predicted similar bond lengths for the scandium dimer. Based on these studies and the results of our investigation, we predict a bond length in the range of 2.53– 2.68 Å for Sc2. The bond length of Ti2 has been well determined in a resonant two-photon ionization experiment to be 1.9429± 0.0013 Å by Doverstål et al.52 In several theoretical works46,65–67 this molecule was studied, too. From a LDA calculation with relativistic effective core potentials, Lu et al.65 estimated a bond length in close agreement with the experimental value. Different CASSCF studies66,67 found very similar bond lengths of 1.994 and 1.995 Å for this molecule. Barden et al.46 found bond lengths between 1.861 and 1.950 Å applying different DFT functionals. Our calculated


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bond lengths with GGA basis sets in combination with GGA functionals are in good agreement with the experimental value 共Table II兲. Spain et al.53 determined the bond length of the vanadium dimer from the resonant two-photon ionization spectrum of 51V2 to be 1.77 Å. Several DFT studies of the vanadium dimer17,46,65,68 report bond distances of around 1.77 Å. To the best of our knowledge, the only wave function calculation on this dimer was reported by Walch et al.69 Also in this work a bond length of 1.77 Å was found. Our calculations show that the V2 bond lengths from local VWN calculations are in excellent agreement with experiment. The GGA functionals overestimate the bond length. This defect is reduced by the GGA optimized basis sets. The bond length decreases systematically with basis set size as Table II shows. Bondybey and English produced Cr2 in a pulsed yttrium aluminum garnet laser vaporization source.54 From the analysis of the fluorescence spectra of Cr2, they obtained an equilibrium bond length of 1.6788 Å. Several theoretical works have been performed at DFT level in order to determine the bond length for this molecule,44–47 too. Independently from the basis set, we obtain excellent agreement with experiment using GGA functionals. The only experimental estimate for Mn2 gives a nonprecise and quite long bond distance of around 3.4 Å.55 Different DFT calculations have been performed using several GGA functionals.44,46,48–50 As Table II shows the Mn2 bond length varies in our calculations between 2.6 and 2.75 Å and, therefore, is considerably shorter than the experimentally estimated upper bound. Purdum et al. used rare gas matrix isolation techniques in combination with extended x-ray absorption fine structure spectroscopy to study the variation in interatomic distances of small Fe molecules.56 From these studies they determined the equilibrium distance of Fe2 to be 2.02± 0.02 Å. DFT studies44–47 were carried out for this dimer, too. Our optimizations of Fe2 with the DZVP and TZVP GGA basis sets in combination with the PW86 functional yield bond distances of 2.033 and 2.008 Å, respectively. These values are in very good agreement with experiment. Unfortunately, no experimental bond length is available for Co2. In most theoretical works,44,45,70,71 a bond length of around 2 Å is suggested for the cobalt dimer. Considerably longer bond lengths 共2.36– 2.56 Å兲 are only found in a configuration interaction study72 and a more recent DFT study.46 Our symmetry broken optimizations predict bond lengths between 2.20 and 2.25 Å. If the symmetry is not broken we obtain bond lengths of around 2 Å as in most other DFT studies. The equilibrium bond distance of Ni2 is one of the most precisely determined bond lengths of all 3d TM dimers. Pinegar et al. used resonant two-photon ionization spectroscopy to study jet-cooled Ni2 produced by a pulsed laser ablation of a nickel target in the throat of a supersonic nozzle using argon as carrier gas. They determined an equilibrium bond distance of 2.1545± 0.0004 Å.57 DFT studies of Ni2 with different GGA functionals44–47 have also been performed. As

Table II shows our GGA calculations show good agreement with experiment, independently of the basis set. Similar to the nickel dimer, a very accurate experimental equilibrium bond distance exists for Cu2. Ram et al. determined the bond length in the copper dimer to be 2.2193± 0.0003 Å from a gas phase Fourier transform emission spectrum.58 The theoretical literature about this dimer is also quite rich.44–47,73,74 Our calculated VWN values are in excellent agreement with the reported experimental bond length. The GGA optimized bond lengths are usually too long. Only with the TZVP-GGA basis in combination with the PW86 functional that a bond length in fair agreement with experiment is obtained. Table II shows that the DZVP-GGA/VWN bond length are for all 3d TM dimers too short, whereas the corresponding PW86 bond lengths are too long. Thus, the experimental bond lengths are bracketed by these two methods. The difference between the two different GGA functionals, BLYP and PW86, is in all cases rather small. This demonstrates the transferability of the newly optimized GGA basis sets. Overall, the best results are obtained with the TZVP-GGA/PW86 method. The optimized bond length from this theoretical approach differs usually by 2 to 3 pm from experiment. This error is only slightly larger than for main group elements. C. Harmonic frequencies

The calculated harmonic vibrational frequencies for the 3d TM dimers are listed in Table III, together with the available experimental data.58,75–82 The theoretical harmonic vibrational frequencies indicate that the reported ground states for the studied TM dimers are minima on the potential energy surface, since no imaginary frequencies were found. Only for the manganese dimers experimental frequencies are not available. Therefore, vibrational frequencies are a valuable source for the validation of theoretical methods. However, large anharmonicity effects may occur in TM dimers which can compromise the direct comparison between the calculated harmonic frequencies and the experimental frequencies. The comparison of calculated frequencies from different theoretical methods in Table III reveals that basis set effects are negligible. The difference between LDA and GGA frequencies is usually below 50 cm−1. An exception represents Co2 where the VWN frequencies are much larger than the corresponding GGA frequencies. In this particular case the GGA calculations yield symmetry broken solutions that are in much better agreement with experiment than the D⬁h VWN solutions. The comparison with the experimental data shows good agreement for the early 共Sc2 and Ti2兲 and late 共Co2, Ni2, and Cu2兲 3d TM dimers. The huge discrepancy between experimental and theoretical frequencies for V2, Cr2, and Fe2 is most likely due to large anharmonicities in these systems. D. Dissociation energies

In Table IV the calculated dissociation energies of the 3d TM dimers are listed, together with the corresponding available experimental data.2,83–90 For Cr2 we have listed in Table IV the most recent experimental dissociation energy.86 An


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TABLE III. Calculated harmonic vibrational frequencies 共in cm−1兲 of 3d transition metal dimers. The available experimental data are reported for comparison, too. Sc2 ⌺u

5

Ti2 ⌬g

3

V2 ⌺g

3

Cr2 ⌺g

Mn2 11 ⌺g

1

Fe2 ⌬u

Co2 5 ⌬g

7

Ni2 ⌺g

Cu2 1 ⌺g

3

DZVP-LDA

VWN BLYP PW86

225.7 209.0 247.1

490.8 446.4 449.1

683.7 627.4 632.6

814.9 752.4 747.2

211.4 177.0 198.4

449.6 397.7 409.5

425.1 256.0a 270.8a

336.3 285.6 299.4

268.8 232.3 246.8

DZVP-GGA

VWN BLYP PW86

215.3 245.8 252.3

475.3 417.3 431.8

653.4 605.4 606.0

809.5 745.8 741.5

216.1 184.6 203.4

432.7 386.3 399.3

411.2 249.0a 290.3a

334.5 291.3 299.8

268.2 234.0 245.7

TZVP-GGA

VWN BLYP PW86

216.8 239.0 254.1

501.2 439.5 438.0

684.5 626.0 626.8

805.2 739.8 736.8

227.1 191.2 209.3

450.2 400.7 413.6

354.6 258.1a 273.4a

357.7 316.8 328.7

283.8 241.0 255.0

239.9b

407.9c

537.5c

480.6d

¯

299.7e

296.8f

259.2g

266.4h

Expt. a

Symmetry broken solution. Reference 75. c Reference 76. d Reference 77.

e

b

f

older literature source91 reports an experimental dissociation energy of 1.44 eV for this dimer. Except for the scandium, titanium, manganese, and cobalt dimers, the reported experimental dissociation energies are accurate within 0.1 eV. The accuracy of the scandium, titanium, and cobalt dimer experimental dissociation energies can be estimated to be around 0.25 eV. For the manganese dimer only an upper bound for the experimental dissociation energy is given in the literature. Our calculated dissociation energies include zero point energy corrections. Table IV shows that the dissociation energies of almost all 3d TM dimers improve using GGA optimized basis sets. Most remarkable is the improvement going from DZVPLDA/PW86 to DZVP-GGA/PW86. Only for the scandium and copper dimers no improvement is observed in this case. For several systems, such as Ti2, V2, Fe2, Co2, and Ni2, the basis set change from DZVP-LDA to DZVP-GGA reduces,

and thus improves, the dissociation energy by 1 eV or more. The reason for this improvement is the better representation of the atomic ground state and an improved description of the sd hybridization in the dimers. This indicates that the here described basis set optimization mainly influences the s – d splitting. In fact, main group elements are little affected by the GGA optimization of the contraction coefficients.92 This is also the reason why the copper dimer DZVP-LDA and DZVP-GGA dissociation energies are almost identical. In this dimer the bond is mainly formed by the atomic 4s orbitals and the s – d splitting has little influence on the bonding situation. The scandium dimer situation is less straightforward to explain. In this case the GGA dissociation energies with the DZVP-LDA basis set are already smaller than the suggested experimental value. Here one has to take into account that the experimental value results from a third-law determination that necessarily depends on assumptions about

Reference 78. Reference 81. g Reference 82. h Reference 58.

TABLE IV. Calculated dissociation energies 共in eV兲 of 3d transition metal dimers. The available experimental data are reported for comparison, too. Sc2 ⌺u

5

Ti2 ⌬g

3

V2 ⌺g

3

Cr2 ⌺g

1

Mn2 11 ⌺g

Fe2 ⌬u

Co2 5 ⌬g

7

Ni2 ⌺g

Cu2 1 ⌺g

3

DZVP-LDA

VWN BLYP PW86

2.36 1.27 1.51

4.42 2.66 2.81

5.96 3.73 3.85

2.28 1.12 1.15

1.91 1.28 1.46

4.79 3.36 3.30

4.85 3.39a 3.35a

5.66 3.86 3.79

2.61 2.12 2.28

DZVP-GGA

VWN BLYP PW86

2.00 0.93 1.19

3.42 1.65 1.83

4.75 2.55 2.73

2.29 1.24 1.16

0.96 0.98 0.59

3.36 1.91 1.96

3.07 1.74a 1.74a

3.87 2.08 2.06

2.64 2.14 2.30

TZVP-GGA

VWN BLYP PW86

2.08 0.98 1.27

3.95 2.08 2.30

5.44 3.10 3.33

2.44 1.36 1.29

1.25 0.63 0.87

4.00 2.44 2.54

4.05 1.90a 1.98a

4.75 2.89 2.92

2.72 2.20 2.36

1.65b

1.54c

2.75d

1.53e

1.15g

1.69h

2.07i

2.08j

Expt. a

f

b

g

Symmetry broken solution. Reference 83. c Reference 84. d Reference 85. e Reference 86.

艋0.8f

Reference 2. Reference 87. h Reference 88. i Reference 89. j Reference 90.


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FIG. 2. Mean deviations of optimized bond lengths and angles of the TM1 test set molecules from experimental structure data.

low-lying electronic states which may not be valid. On the other hand, however, the optimized bond lengths of the scandium dimer 共see Table II兲 suggest that van der Waals interaction may not be a negligible contribution in this system. In fact, the similarity in the bond length with the manganese dimer supports this viewpoint. The splitting of the DZVP-GGA basis set to triple zeta shows no further improvement in the dissociation energies of the 3d TM dimers. Instead the already too large dissociation energies of several dimers are further enlarged. As a result the overall quality of the 3d TM dimer dissociation energies deteriorates with the TZVP-GGA basis set. From our experience the limited number of primitive Gaussians in the TZVP basis sets is the main reason for this behavior. VI. MOLECULES

In order to test the performance of the newly developed DZVP-GGA basis set for structure optimization, 78 different TM metal compounds containing 3d elements were optimized with different functionals. These molecules can be grouped in the following classes. Transition metal dimers 共Sc2, Ti2, V2, Cr2, Mn2, Fe2, Co2, Ni2, and Cu2兲, transition metal oxides and hydrides 共ScO, TiO, VO, CrO, CrO3, MnO, FeO, CoO, CuO, CrH, MnH, CoH, NiH, CuH, and ZnH兲, transition metal halogenides 共ScF, CuF, MnF2, CoF2, CuF2, ZnF2, ScF3, FeF3, TiF4, VF5, ScCl, TiCl, FeCl, CoCl, CuCl, MnCl2, FeCl2, CoCl2, ZnCl2, NiCl2, FeCl3, CoCl3, Cu3Cl3, and TiCl4兲, transition metal carbonyls 关V共CO兲6, Cr共CO兲6, MnH共CO兲5, Mn2共CO兲10, Fe共CO兲5, CoH共CO兲4, Ni共CO兲4, VCp共CO兲4, CrBz共CO兲3, MnCp共CO兲3, Fe共C2H4兲共CO兲4, Fe共C4H4兲共CO兲3, and CoCp共CO兲2兴, organometallic compounds 关Ti共CH3兲4, TiCl3CH3, TiCp2Cl2, CrBz2, CrCp2, MnCp2, FeCp2, NiCp2, NiCpNO, Zn共CH3兲2, and ZnCpCH3兴, and mixed compounds 共VOF3, VOCl3, CrO2F2, CrO2Cl2, MnFO3, and CuOH兲. For all systems in this TM1 test set, experimental gas phase structure parameters are available.93–109 Therefore, reliable comparisons between optimized and experimental structure parameters are possible. In Fig. 2 the mean average deviations of the optimized bond lengths and angles from the experimental data for the molecules of the TM1 test set are depicted. The clear histograms refer to the bond lengths, whereas the darker histograms refer to the bond angles. The deviations are given in picometers

and degrees for the bond lengths and angles, respectively. The molecules were optimized with the VWN, BLYP, and PW86 functionals in combination with the newly developed DZVP-GGA basis sets. For the calculation of the exchangecorrelation potential in these optimizations, the BASIS and AUXIS approaches were employed. As Fig. 2 shows the mean average bond distance deviations range between 2.5 and 3 pm. This is roughly twice as large as for bonds between first row elements. The average deviation of bond angles is well below 2°. This is comparable to the optimized bond angle quality of main group element molecules. The comparison of the BASIS and AUXIS approaches shows only small differences in the quality of the optimized structure parameters. Therefore, the DZVP-GGA basis set is compatible with the AUXIS approach, which is particularly important for large scale applications. So far we have discussed the average deviations of the optimized structure parameters of the molecules from the TM1 test set. As already mentioned for the 3d TM dimers, the experimental bond lengths are usually bracketed by the DZVP-GGA/VWN and DZVP-GGA/PW86 optimized bond lengths. This observation holds for most other systems in the TM1 test set, too. As an example, the 共BASIS兲 optimized DZVP-GGA/VWN 共⫹兲 and DZVP-GGA/PW86 共䊏兲 TM-CO bond lengths are plotted against the corresponding experimental values for the transition metal carbonyls of the TM1 set in Fig. 3. This figure shows that the bracketing rule holds for almost all carbonyls in the test set. Small violations of this rule are observed for Ni共CO兲4 and Mn2共CO兲10. In these cases the DZVP-GGA/PW86 optimized TM-CO bonds are already slightly shorter than the corresponding experimental value. Only one system, CoCp共CO兲2, shows a large violation of the bracketing rule. In this case the DZVP-GGA/VWN optimized TM-CO bonds are longer than the reported experimental bond lengths. This usually indicates a problem with the experimental data. The here described bracketing of the experimental bond length by the DZVP-GGA/VWN and DZVP-GGA/PW86 optimized bond lengths holds also for other bonds between transition metals and main group elements. For the TM-CO bond Fig. 3 shows that the optimized DZVP-GGA/PW86


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FIG. 3. Comparison of theoretical and experimental transition metal CO distances in the transition metal carbonyls of the TM1 test set. The abbreviations Bz and Cp refer to benzene and cyclopentadienyl groups, respectively.

bond lengths are in most cases close to the experimental values. This is a more general trend if GGA optimized basis sets are used.

VII. CONCLUSIONS

The GGA optimized DZVP and TZVP basis sets for 3d transition metals are presented. It is shown that the s2dn−1 → s1dn excitation energies of the 3d transition metal atoms considerably improve if the DZVP-GGA basis set in combination with the PW86 functional is used. As a result, all ground states of the 3d transition metal atoms are correctly predicted at this level of theory. The optimization of the 3d transition metal dimers with the DZVP-GGA/PW86 method yields bond lengths that are in good agreement with the available experimental data. Compared to the DZVP-LDA/PW86 level of theory, no significant changes are observed. The only exception is the cobalt dimer where the symmetry broken solution was easier obtained with the DZVP-GGA basis set. The analysis of the optimized structure parameters of the TM1 test set reveals a very similar conclusion, namely, that the DZVP-GGA/PW86 optimized structure parameters are quite reliable for transition metal bonds and angles. This analysis also shows that the use of the auxiliary function density for the calculation of the exchange-correlation potential has almost no influence on the quality of the optimized structure parameters. Significant differences between the LDA and GGA optimized basis sets arise, however, for the atomization energies. As an example, we analyzed in detail the dissociation energies of the 3d transition metal dimers. With the DZVP-GGA/ PW86 method the error in the dissociation energies drops below 0.5 eV for most dimers. This represents a considerable improvement compared with the DZVP-LDA/PW86 method. It should be noted that this improvement comes without any additional computational demand. This strongly encourages the use of GGA optimized basis sets.

The analysis of the 3d transition metal dimer calculations also shows that the GGA optimized basis sets are transferable between different GGA functionals. Much less satisfying are, however, the results from the GGA optimized TZVP basis set. Our analysis indicates that the limited amount of primitive Gaussians in this basis set might be the reason for the relative poor performance. Therefore, it is desirable to repeat the GGA basis set optimization here described with a denser spaced basis set optimized for DFT calculations. Work in this direction is currently underway in our laboratory. ACKNOWLEDGMENT

This work was financially supported by the CONACYT Project Nos. 36037-E, 40379-F, and U48775. APPENDIX: AUTOMATIC GENERATION OF AUXILIARY FUNCTIONS

With the auxiliary function definitions GEN-An and GEN-An*, where n ranges from 2 to 4, automatically generated auxiliary function sets are selected in deMon2k. The GEN-An sets consist of s, p, and d Hermite Gaussian functions. The GEN-An* sets also possess f and g Hermite Gaussians. Because the auxiliary functions are used to fit the electronic density they are grouped in s, spd, and spdfg sets. The exponents are shared within each of these sets.110,111 The auxiliary function notation 共3,2,2兲 describes three s sets together with three functions, two spd sets together with 20 functions, and two spdfg sets together with 70 functions. The exponent range of the auxiliary functions is determined by the smallest, ␨min, and largest, ␨max, primitive Gaussian exponents of the chosen basis set. Therefore, the GEN-An and GEN-An* automatically generated auxiliary function sets are different for different basis sets. The number of exponents N 共auxiliary function sets兲 is given by


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冉

N = Int

冊

ln共␨max/␨min兲 + 0.5 . ln共6 − n兲

18

共A1兲

Here n is 2, 3, or 4 according to the chosen GEN-An or GEN-An* set. The exponents are generated almost, as it is explained below, even tempered and split into s, spd, and, if a GEN-An* set is requested, spdfg sets. The tightest 共largest兲 exponents are assigned to the s sets, followed by the spd and, if exist, spdfg sets. The basic exponent from which the generation starts is defined as

␨o = 2␨min共6 − n兲共N−1兲 .

共A2兲

From this ␨o exponent the two tightest s set exponents, ␨1 and ␨2, are generated according to the formulas

冉

␨1 = 1 +

␨2 =

冊

n ␨o , 12 − 2n

␨o . 6−n

共A3兲

共A4兲

The other s set exponents are generated according to the even tempered progression:

␨i+1 =

␨i . 6−n

共A5兲

The ␨o exponent of the following spd sets is also generated according to the progression 共A5兲. Based on this ␨o the exponents of the first two spd sets are calculated by formulas 共A3兲 and 共A4兲. The following spd set exponents are then calculated again according to the even tempered progression 共A5兲. In the same way the spdfg set exponents are calculated. In the case of 3d elements an extra diffuse s auxiliary function is added. W. Weltner and R. J. Van Zee, Annu. Rev. Phys. Chem. 35, 291 共1984兲. M. D. Morse, Chem. Rev. 共Washington, D.C.兲 86, 1049 共1986兲. 3 M. Moskovits, Metal Clusters 共Wiley, New York, 1986兲. 4 M. A. Duncan, Advances in Metal and Semiconductor Clusters 共JAI, Greenwich, CT, 1993兲. 5 D. R. Salahub, Adv. Chem. Phys. 69, 447 共1987兲. 6 J. A. Alonso, Chem. Rev. 共Washington, D.C.兲 100, 637 共2000兲. 7 S. N. Khanna, C. Ashman, B. K. Rao, and P. Jena, J. Chem. Phys. 114, 9792 共2001兲. 8 J. R. Lombardi and B. Davis, Chem. Rev. 共Washington, D.C.兲 102, 2431 共2002兲. 9 R. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules 共Oxford University Press, New York, 1989兲. 10 R. M. Dreizler and E. K. U. Gross, Density Functional Theory 共Springer, Berlin, 1990兲. 11 J. Labanowski and J. Andzelm, Density Functional Methods in Chemistry 共Springer, Berlin, 1991兲. 12 J. M. Seminario and P. Politzer, Density Functional Theory: A Tool for Chemistry 共Elsevier, Amsterdam, 1995兲. 13 W. Kohn, Rev. Mod. Phys. 71, 1253 共1999兲. 14 D. R. Salahub, M. Castro, R. Fournier et al., in Theoretical and Computational Approaches to Interface Phenomena, edited by H. Seller and J. J. Golad 共Plenum, New York, 1994兲, pp. 187–218. 15 D. R. Salahub, J. Weber, A. Goursot, A. M. Köster, and A. Vela, in Theory and Applications of the Computational Chemistry: The First 40 Years, edited by C. E. Dykstra, G. Frenking, K. S. Kim, and G. Scuseria 共Elsevier, Amsterdam, 2005兲. 16 T. Mineva, A. Goursot, and C. Daul, Chem. Phys. Lett. 350, 147 共2001兲. 17 P. Calaminici, A. M. Köster, N. Russo, P. N. Roy, T. Carrington, Jr., and D. R. Salahub, J. Chem. Phys. 114, 4036 共2001兲. 1 2

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