Introduction to Electronic Structure Theory

Computational Chemistry: Molecular Simulations with Chemical and Biological Applications Prof. M. Meuwly, Dr. PA Cazade, Dr. M.-W. Lee

Overview 1. Electronic Structure of Molecules 1.1 The Electronic Problem 1.2 The Hartree Fock Equations 1.3 Basis Sets 1.4 Solving the HF equations 1.5 Correlated Methods 1.6 Density Functional Theory 2. Molecular Simulations 2.1 Force Fields and Energy Functions 2.2 Molecular Dynamics Simulations (MD) 2.3 Analysis of MD Simulations 2.4 Monte Carlo Simulations (MC) 2.5 Free Energy Simulations

Overview Literature: Electronic Structure: • Quantum Chemistry by Szabo/Ostlund (Dover) • Quantum Chemistry by Ira Levine (Pearson) Molecular Simulations: • Molecular Modelling by A. Leach (Prentice Hall) • Understanding Molecular Simulation by Frenkel and Smit (AP) • Computer Simulation of Liquids by Allen and Tildesley (OUP) Credits: • Exercises • 10 minute presentation

Overview Exercises: • Throughout the semester – see semester plan • Applied examples with Gaussian09 and CHARMM Hours: Mon Thu

16-17 15-17

Overview How well do we need to describe intermolecular interactions in order to contribute to interpretation, understanding and prediction of chemical processes? Depending on the observable in question, what level of detail is required? What can we learn about intermolecular interactions from comparing simulation results with experiments?

“Small Molecules” Example: Bent versus Linear Methylene 1959 Herzberg and Shoosmith (Nature, 1959, Exp) conclude it is linear 1960 Foster and Boys (J. Chem. Phys., 1960, Comp) predict an angle of 128o 1970 Bender and Schaefer III (J. Am. Chem. Soc., 1970, Comp) confirm bent structure (135o) 1971 Herzberg and Johns (J. Chem. Phys., 1971) reinterpret spectra and confirm bent structure. Currently accepted value is 135.5o (K Kuchitsu (ed) "Structure of Free Polyatomic Molecules - Basic Data" Springer, Berlin, 1998) Such structure determination relies nowadays on fitting spectroscopic data to a (model) Hamiltonian. “Large Molecules” 1960 Perutz and coworkers Structure of haemoglobin – 3-dimensional Fourier synthesis at 5.5-A resolution, obtained by X-ray analysis Nature, 1960, 416 (1960) 1994 Schlichting and coworkers: Crystal Structure of Photolyzed Carbonmonoxy-Myoglobin Nature, 808 (1994) 2003 Anfinrud and coworkers: Watching a protein as it functions with 150-ps time-resolved X-ray crystallography Science, 300 (2003)

Introduction to Electronic Structure Calculations The Hartree Fock Equations

Molecular properties

Transition States Reaction coords.

Ab initio electronic structure theory Hartree-Fock (HF) Electron Correlation (MP2, CI, CC, etc.)

Geometry prediction

Benchmarks for parametrization

Spectroscopic observables

Assist Experimentalists

Goal: Insight into chemical phenomena.

Computational Chemistry

Rationalizing a strained allyl structure Usually cis but trans because of strain through binding to Pd

Computational Chemistry

“organic splitting” of H2

The Problem What is a molecule? A molecule is “composed” of atoms, or, more generally a collection of charged particles, positive nuclei and negative electrons. The interaction between charged particles is described by;

qi q j qiq j = Vij = V (rij ) = rij 4 πε0 rij Coulomb Potential

rij

qj

qi

Coulomb interaction between these charged particles is the only important physical force necessary to describe chemical phenomena.

But, electrons and nuclei are in constant motion In Classical Mechanics, the dynamics of a system (i.e. how the system evolves in time) is described by Newton’s 2nd Law: F = force F = ma a = acceleration 2 dV d r r = position vector − =m 2 dr dt m = particle mass

In Quantum Mechanics, particle behavior is described in terms of a wavefunction, Ψ. Time-dependent Schrödinger Equation

∂Ψ ˆ HΨ = i ∂t  

Hˆ

(

i = −1;= h 2π  

)

Hamiltonian Operator

Time-Independent Schrödinger Equation Hˆ (r,t) = Hˆ (r) −iEt /  Ψ(r,t) = Ψ(r)e  

Hˆ (r)Ψ(r) = EΨ(r)

If H is time-independent, the timedependence of Y may be separated out as a simple phase factor.

Time-Independent Schrödinger Equation

Describes the stationary properties of electrons.

Hamiltonian for a system with N-particles Hˆ = Tˆ + Vˆ

Sum of kinetic (T) and potential (V) energy

N 2 2  2 2 2    ∂ ∂ ∂ 2 Tˆ = ∑ Tˆi = −∑ ∇ i = −∑  2 + 2 + 2 2m 2m ∂y i ∂zi  i i  ∂x i i=1 i=1   i=1 N

N

 ∂2 ∂2 ∂2  ∇ = 2 + 2 + 2  ∂x i ∂y i ∂zi  2 i

N

N

N

Kinetic energy

Laplacian operator

N

qi q j ˆ V = ∑ ∑Vij = ∑ ∑ i=1 j>1 rij i=1 j>1

Potential energy

When these expressions are used in the time-independent Schrodinger Equation, the dynamics of all electrons and nuclei in a molecule or atom are taken into account.

Born-Oppenheimer Approximation • So far, the Hamiltonian contains the following terms: +

Hˆ = Tˆn + Tˆe + Vˆne + Vˆee + Vˆnn Tˆn Tˆe Vˆne Vˆee Vˆnn •

Since nuclei are much heavier than electrons, their velocities are much smaller. To a good approximation, the Schrödinger equation can be separated into two parts: – One part describes the electronic wavefunction for a fixed nuclear geometry. – The second describes the nuclear wavefunction, where the electronic energy plays the role of a potential energy.

Born-Oppenheimer Approx. cont.

•

In other words, the kinetic energy of the nuclei can be treated separately. This is the Born-Oppenheimer approximation. As a result, the electronic wavefunction depends only on the positions of the nuclei.

•

Physically, this implies that the nuclei move on a potential energy surface (PES), which are solutions to the electronic Schrödinger equation. Under the BO approx., the PES is independent of the nuclear masses; that is, it is the same for isotopic molecules.

.

E

.

H + H 0

H •

H

Solution of the nuclear wavefunction leads to physically meaningful quantities such as molecular vibrations and rotations.

Limitations of the Born-Oppenheimer approximation • The total wavefunction is limited to one electronic surface, i.e. a particular electronic state. • The BO approx. is usually very good, but breaks down when two (or more) electronic states are close in energy at particular nuclear geometries. In such situations, a “ non-adiabatic” wavefunction - a product of nuclear and electronic wavefunctions - must be used. • In writing the Hamiltonian as a sum of electron kinetic and potential energy terms, relativistic effects have been ignored. These are normally negligible for lighter elements (Zi

Hˆ e = ∑ hˆi + ∑ ∑ gˆij + Vˆnn

Total Hamiltonian

Calculation of the energy.

E e = Φ | Hˆ e | Φ N −1

E e = Aˆ Π | Hˆ e | Aˆ Π = ∑ (−1) p Π | Hˆ e | Pˆ Π

Expectation value over Slater Determinant

p= 0

Examine specific integrals:

Φ | Vˆnn | Φ = Vnn

Nuclear repulsion does not depend on electron coordinates.

For coordinate 1, Π | hˆ1 | Π = [φ1 (1)φ 2 (2)φ N (N)]| hˆ1 | [φ1 (1)φ 2 (2)φ N (N)]

 

= φ1 (1) | hˆ1 | φ1 (1) φ 2 (2) | φ 2 (2)  φN (N ) | φN (N ) = h1 The one-electron operator acts only on electron 1 and yields an energy, h1, that depends only on the kinetic energy and attraction to all nuclei.

Π | gˆ12 | Π = [φ1 (1)φ 2 (2)φ N (N)]| gˆ12 | [φ1(1)φ 2 (2)φ N (N )] = φ1 (1)φ2 (2) | gˆ12 | φ1 (1)φ2 (2) φ 3 (3) | φ 3 (3)  φN (N ) | φ N (N )  

= φ1 (1)φ2 (2) | gˆ12 | φ1 (1)φ2 (2) = J12 Coulomb integral, J12: represents the classical repulsion between two charge distributions φ12(1) and φ22(2).

Π | gˆ12 | Pˆ12Π = [φ1 (1)φ 2 (2)φ N (N)]| gˆ12 | [φ 2 (1)φ1 (2)φ N (N)]

= φ1 (1)φ2 (2) | gˆ12 | φ2 (1)φ1 (2) φ 3 (3) | φ 3 (3)  φN (N ) | φ N (N )  

= φ1 (1)φ2 (2) | gˆ12 | φ2 (1)φ1 (2) = K12

Exchange integral, K12: no classical analogue. Responsible for chemical bonds.

The expression for the energy can now be written as:

1 N N E e = ∑ hi + ∑ ∑ (J ij − Kij ) + Vnn 2 i j i =1 N

Sum of one-electron, Coulomb, and exchange integrals, and Vnn.

To apply the variational principle, the Coulomb and Exchange integrals are written as operators, N

Ee = ∑ i =1

(

N N 1 φ i | hˆi | φ i + ∑ ∑ φ j | Jˆi | φ j − φ j | Kˆ i | φ j 2 i j

)+ V

nn

Jˆi | φ j (2) = φ i (1) | gˆ12 | φ i (1) φ j (2) Kˆ i | φ j (2) = φ i (1) | gˆ12 | φ j (1) φ i (2) The objective now is to find the best orbitals (φi, MOs) that minimize the energy (or at least remain stationary with respect to further changes in φi), while maintaining orthonormality of φi.

• Employ the method of Langrange Multipliers: Function to optimize. f (x1 ,x 2 ,x N ) Rewrite in terms of another function. g(x1 ,x 2 ,x N ) = 0 Define Lagrange L(x1 ,x 2 ,x N ,λ ) = f (x1, x 2 , x N ) − λ g(x1 ,x 2 ,x N ) function. ∂L ∂L = 0, = 0 Constrained optimization of L. Optimize L such that ∂ ∂x λ i i   • In terms of molecular orbitals, the Langrange function is:

(

N

L = E − ∑ λ ij φ i | φ j − δ ij

)

Constraint is orthogonality of the φi

ij N

(

δL = δE − ∑ λ ij δφ i | φ j + φi | δφ j

)

Change in L with respect to small changes in φi should be zero.

=0

ij

• Change in the energy with respect changes in φi. N

δE = ∑ i =1

(

δφ i | hˆi | φ i + φ i | hˆi | δφi

)+ ∑ (δφ | Jˆ − Kˆ N

i

ij

j

j

| φ i + φi | Jˆ j − Kˆ j | δφ i

)

Define the Fock Operator, Fi N

(

Fˆi = hˆi + ∑ Jˆ j − Kˆ j j N

)

Effective one-electron operator, associated with the variation in the energy.

(

δE = ∑ δφ i | Fˆi | φi + φ i | Fˆi | δφi i =1

N

δL = ∑ i=1

(

)

)

Change in energy in terms of the Fock operator. N

(

δφ i | Fˆi | φ i + φ i | Fˆi | δφ i − ∑ λ ij δφ i | φ j + φ i | δφ j

)= 0

ij

According to the variational principle, the best orbitals, φi, will make δL=0. After some algebra, the final expression becomes: N

Fˆiφi = ∑ λ ijφ j j

Hartree-Fock Equations

After a unitary transformation, λij→0 and λii→εi.

Fˆiφi '= ε iφi ' εi = φ i '| Fˆi | φ i '

HF equations in terms of Canonical MOs and diagonal Lagrange multipliers.

Lagrange multipliers can be interpreted as MO energies.

Note: 1. The HF equations cast in this way, form a set of pseudo-eigenvalue equations. 2. A specific Fock orbital can only be determined once all the other occupied orbitals are known. 3. The HF equations are solved iteratively. Guess, calculate the energy, improve the guess, recalculate, etc. 4. A set of orbitals that is a solution to the HF equations are called Self-consistent Field (SCF) orbitals. 5. The Canonical MOs are a convenient set of functions to use in the variational procedure, but they are not unique from the standpoint of calculating the energy.

Basis Set Approximation •

In most molecular calculations, the unknown MOs are expressed in terms of a known set of functions - a basis set. Two criteria for selecting basis functions. I) They should be physically meaningful. ii) computation of the integrals should be tractable.

•

•

It is common practice to use a linear expansion of Gaussian functions in the MO basis because they are easy to handle computationally.

Each MO is expanded in a set of basis functions centered at the nuclei and are commonly called Atomic Orbitals. (Molecular orbital = Linear Combination of Atomic Orbitals - LCAO).

MO Expansion M

φi = ∑ cαi χα α

M

M

Fˆi ∑ cα i χα = ε i ∑ cαi χ α α

LCAO - MO representation Coefficients are variational parameters

HF equations in the AO basis

α

FC = SCε Fαβ = χα | Fˆ | χ β Sαβ = χ α | χ β

Matrix representation of HF eqns. Roothaan-Hall equations (closed shell) Fαβ - element of the Fock matrix Sαβ - overlap of two AOs

Roothaan-Hall equations generate M molecular orbitals from M basis functions. • N-occupied MOs • M-N virtual or unoccupied MOs (no physical interpretation)

Total Energy in MO basis

(

N N 1 φ i | hˆi | φ i + ∑ ∑ φ iφ j | gˆ | φ iφ j − φ iφ j | gˆ | φ jφ i 2 i j

N

E=∑ i =1

)+ V

nn

Total Energy in AO basis N

M

E = ∑ ∑ cα ic βi i =1 αβ

N

(

M

1 χα | hˆi | χ β + ∑ ∑ cαic γj c βicδj χα χ γ | gˆ | χ β χδ − χα χ γ | gˆ | χδ χ β 2 ij αβγδ

One-electron integrals, M2

Dγδ =

∑ j

c γj cδj ;

Dαβ =

nn

Two-electron integrals, M4 Computed at the start; do not change

Products of AO coeff form Density Matrix, D occ.MO

)+ V

occ.MO

∑ i

cα ic βi

General SCF Procedure

Obtain initial guess for coeff., cαi,form the initial Dγδ

Form the Fock matrix

Iterate Diagonalize the Fock Matrix

Form new Density Matrix

Two-electron integrals

Computational Effort • Formally, the SCF procedure scales as M4 (the number of basis functions to the 4th power).

Accuracy • As the number of functions increases, the accuracy of the Molecular Orbitals improves. • As M→∞, the complete basis set limit is reached ⇒ Hartree-Fock limit. • Result: The best single determinant wavefunction that can be obtained. (This is not the exact solution to the Schrodinger equation.)

Practical Limitation • In practice, a finite basis set is used; the HF limit is never reached. • The term “Hartree-Fock” is often used to describe SCF calculations with incomplete basis sets.

Restricted and Unrestricted Hartree-Fock Restricted Hartree-Fock (RHF) For even electron, closed-shell singlet states, electrons in a given MO with α and β spin are constrained to have the same spatial dependence. Restricted Open-shell Hartree-Fock (ROHF) The spatial part of the doubly occupied orbitals are restricted to be the same. Unrestricted Hartree-fock (UHF) α and β spinorbitals have different spatial parts. α 5 4

Energy

3 2 1

RHF Singlet

ROHF Doublet

UHF Doublet

β

}

Spinorbitals

φiσ(n)

Comparison of RHF and UHF R(O)HF • α and β spins have same spatial part

UHF • α and β spins have different spatial parts

• Wavefunction, Φ, is an eigenfunction of S2 operator.

• Wavefunction is not an eigenfunction of S2. Φ may be contaminated with states of higher multiplicity (2S+1).

• For open-shell systems, the unpaired electron (α) interacts differently with α and β spins. The optimum spatial orbitals are different. ∴ Restricted formalism is not suitable for spin dependent properties. • Starting point for more advanced calculations that include electron correlation. • Can not describe dissociation appropriately

• EUHF ≤ ER(O)HF • Yields qualitatively correct spin densities. • Starting point for more advanced calculations that include electron correlation. • Correct behaviour at long range.

Ab Initio (latin, “from the beginning”) Quantum Chemistry Summary of approximations • • • • • •

Born-Oppenheimer Approx. Non-relativistic Hamiltonian Use of trial functions, MOs, in the variational procedure Single Slater determinant Basis set, LCAO-MO approx. RHF, ROHF, UHF

Consequence of using a single Slater determinant and the Self-consistent Field equations: Electron-electron repulsion is included as an average effect. The electron repulsion felt by one electron is an average potential field of all the others, assuming that their spatial distribution is represented by orbitals. This is sometimes referred to as the Mean Field Approximation.

Electron correlation has been neglected!!!

Essential points • • • • • •

HF is first order approximation (no correlation) Introduction of a basis set to represent MOs allows practical calculations In practice, convergence of observables with basis set size needs to be established. RHF not suitable for dissociation problems UHF suitable but beware of spin contamination (in particular for metal containing systems) So far single-reference calculations (only one Slater determinant). Probably appropriate for most organic molecules but not necessarily for metalcontaining ones.

Introduction to Electronic Structure Calculations Basis Sets

Basis Set Approximation MOs are expanded in terms of Atomic Orbitals M

φi = ∑ cαi χα α

LCAO - MO representation Coefficients are variational parameters

• φi (MO) is initially unknown; describing (expanding) the MO as a combination of known (χ) AO functions. • As M→∞, reach the complete basis set limit; not an approximation. • When M is finite, the representation is approximate. Two criteria for selecting basis functions. i) They should be physically meaningful. ii) computation of the integrals should be tractable.

Slater Type Orbitals (STO)

χζ ,n,l ,m (r,θ ,ϕ) = NYl,m (θ ,ϕ )r e

n−1 −ζ r

STO depends on quantum numbers n,l,m and zeta, ζ .

Yl,m (θ,ϕ ) Spherical harmonics;

N - normalization

Advantages: 1. Physically, the exponential dependence on distance from the nucleus is very close to the exact hydrogenic orbitals. 2. Ensures fairly rapid convergence with increasing number of functions. Disadvantages: 1. Three and four center integrals cannot be performed analytically. 2. No radial nodes. These can be introduced by making linear combinations of STOs. Practical Use: 1. Calculations of very high accuracy, atomic and diatomic systems. 2. Semi-empirical methods where 3- and 4-center integrals are neglected.

Gaussian Type Orbitals (GTO)

χζ ,n,l ,m (r,θ ,ϕ) = NYl,m (θ ,ϕ )r

2n− 2−l

e

−ζ r 2

Polar coordinates

GTO depends on quantum numbers n,l,m and exponent zeta, ζ. d-function has five components (Y2,2,Y2,1,Y2,0,Y2,-1,Y2,-2).

χζ ,l

x ,ly

,lz

(x,y,z) = Nx

lx

ly

lz

y z r

2n −2− l −ζ r 2

e

Cartesian coordinates

In Cartesian coords., the angular dependence of the GTO is computed from the sum of lx, ly, and lz (lx+ly+lz =1, a p-orbital). • d-function has six components (x2, y2, z2, xy, xz, yz) in cartesian coord. These may be transformed to spherical functions plus one extra s-type function: (x2+y2+z2). • f-orbitals have 10 components, which may be transformed to the 7-‘pure’ spherical ones plus 3 p-type functions.

GTOs are inferior to STOs in three ways: 1. 2. 3.

At the nucleus, the GTO has zero slope; the STO has a cusp. Behavior near the nucleus is poorly represented. GTOs diminish too rapidly with distance. The ‘tail’ behavior is poorly represented. Extra d-, f-, g-, etc. functions (from Cart. rep.)may lead to linear dependence of the basis set. They are usually dropped when large basis sets are used.

Advantage: GTOs have analytical solutions. Use a linear combination of GTOs to overcome these deficiencies.

Classification Minimum basis: Only enough functions are used to contain the the electrons of the neutral atoms (usually core plus valence orbitals). 1st row: 1s, 2s, 2p 2nd row: 1s, 2s, 3s, 2p, 3p

5-AOs 9-AOs

Double Zeta (DZ) basis: Double the number of all basis functions. Hydrogen has two 1s-functions: 1s and 1s´ Li-Ne: 1s and 1s´, 2s and 2s´, 2p and 2p´

2-AOs 10-AOs

Think of 1s and 1s´ as ‘inner’ and ‘outer’ functions. The inner function has larger ζ exponent and is ∴tighter, outer 1s´ has a smaller ζ, more diffuse.

DZ basis yields a better description of the charge distribution compared to a minimal basis. σ Consider HCN,

⇒ ⇒

π

Charge distributions are different in different parts of the molecule.

C-H σ-bond consists of the H 1s orbital and the C 2pz. CN π-bond is made up of C and N 2px (and 2py) AOs. Because the π-bond is more diffuse, the optimal exponent ζ for px (py) should be smaller than that for the more localized pz orbital.

DZ basis has the flexibility (while the minimal basis does not) to describe the charge distribution in both parts of the molecule. The optimized AO coefficient (in MO expansion) of the ‘tighter’ inner pz function on carbon will be larger in the C-H bond. The more diffuse outer px and py functions will have larger AO coefficients in the π-bond.

Split Valence Basis Sets •

Doubling the number of functions provides a much better description of bonding in the valence region.

•

Doubling the number of functions in the core region improves the description of energetically important but chemically uninteresting core electrons.

•

Split valence basis sets improve the flexibility of the valence region and use a single (contracted) set of functions for the core.

VDZ VTZ VQZ V5Z V6Z

double zeta triplet zeta quadruple zeta quintuple zeta sextuple zeta

2x number of basis functions in valence region 3x “ 4x “ 5x “ 6x “

Polarization Functions Consider HCN, σ

π

H-C σ-bond: Electron distribution along the CH bond is different from the perpendicular direction. The H 1s orbital does not describe this behavior well. If p-functions are added to hydrogen, then the pz AO can improve the description of the CH bond.

H 1s

H 2pz

p-functions induce a polarization of s-orbitals. d-function induce polarization of p-orbitals, etc.

For a single determinantal wavefunction, the 1st set of polarization functions is by far the most important and will describe most if not all of the important charge polarization effects.

Polarization Functions cont.

• To describe charge polarization effects at the SCF level, add – P-functions to H (one set) – D-functions to Li-Ne, Na-Ar (one set 1st row, 1-2 sets for 2nd row)

• To recover a larger fraction of the dynamical correlation energy, multiple functions of higher angular momentum (d, f, g, h, i…) are essential. Electron correlation - energy is lowered by electrons avoiding each other. Two types: 1) Radial correlation - two electrons, one close to the nucleus the other farther away. Need basis functions of the same type but different exponent. (tight and diffuse p-functions, for example) 2) Angular correlation - Two electrons on opposite sides of the nucleus. Basis set needs functions with the same exponent but different angular momentum. For s-functions, need p-functions (and d, f, g..) to account for angular correlation. Radial ≈ Angular in importance.

Diffuse Functions Diffuse functions, s-, p-, and d-functions with small exponents are usually added for specific purposes. (1) Calculations on anions. (2) Dipole moment (3) Polarizability

Contracted Basis Sets Energy optimized basis sets have a disadvantage. Many functions go toward representing the energetically important but chemical uninteresting core electrons. Suppose 10s functions have been optimized for carbon. Start with 10 primitive gaussians PGTOs

Inner 6 describe core 1s electrons Next 4 describe valence electrons

End with 3 contracted gaussians CGTOs

Contract to one 1s function contract to two 2s functions

k

χ (CGTO) = ∑ a i χ i (PGTO) i

Energy always increases! Fewer variational parameters. But, less CPU time required.

Pople Style Basis Sets STO-nG

Minimal basis, n=# of gaussian primitives contracted to one STO.

k-nlmG 3-21G

Split valence basis sets** Contraction scheme (6s3p/3s) -> [3s2p/2s] (1st row elements /H) 3 PGTOs contracted to 1, forms core 2PGTOs contracted to 1, forms inner valence 1 PGTO , forms outer valence After contraction of the PGTOs, C has 3s and 2p AOs.

6-31G 6-311G 6-31+G*

(10s4p/4s) -> [3s2p/2s] Valence double zeta basis (11s5p/4s) -> [4s3p/3s] Valence triple zeta basis Equivalent to 6-31+G(d). 6-31G basis augmented with diffuse sp-functions on heavy atoms, polarization function (d) on heavy atoms. 6-311++G(2df,2pd) Triplet split valence; augmented with diffuse sp- on heavy atoms and diffuse s- on H’s. Polarization functions 2d and 1f on heavy atoms; 2p and 1d on H’s. (**In the Pople scheme, s- and p-functions have the exponent. 6-31G(d,p) most common)

Introduction to Electronic Structure Calculations Electron Correlation

What is electron correlation and why do we need it? Φ0 is a single determinantal wavefunction.

Φ SD =  

φ1 (1) φ1 (2)

φ 2 (1)  φ N (1) φ 2 (2)  φ N (2)

    φ1 (N ) φ 2 (N)  φ N (N)

,

φ i | φ j = δij

Slater Determinant Recall that the SCF procedure accounts for electron-electron repulsion by optimizing the one-electron MOs in the presence of an average field of the other electrons. The result is that electrons in the same spatial MO are too close together; their motion is actually correlated (as one moves, the other responds).

Eel.cor. = Eexact - EHF

(B.O. approx; non-relativistic H)

RHF dissociation problem Consider H2 in a minimal basis composed of one atomic 1s orbital on each atom. Two AOs (χ) leads to two MOs (φ)…

H

H

φ2 = N 2 ( χ A − χ B );

antibonding M O

H 1s

H 1s

φ1 = N1 ( χ A + χ B ); H

H

“1” and “2” label the electrons; “A” and “B” the nuclei

bonding M O

The ground state wavefunction is: φ1α (1) φ1β (1) Φ0 = φ1α (2) φ1β (2) Φ 0 = φ1α (1)φ1β (2) − φ1α (2)φ1β (1)

Slater determinant with two electrons in the bonding MO Expand the Slater Determinant

Φ 0 = φ1 (1)φ1 (2)[α (1)β (2) − β (1)α (2)] Factor the spatial and spin parts Only consider spatial part Φ 0 = φ1 (1)φ1 (2) = ( χ A (1) + χ B (1))(χ A (2) + χ B (2)) Four terms in Φ 0 = χ A (1) χ A (2) + χ B (1) χ B (2) + χ A (1) χ B (2) + χ B (1) χ A (2) the AO basis

χA χ A χB χ B χA χ B χB χ A

Ionic terms, two electrons in one Atomic Orbital, i.e. on one H center; H-

Covalent terms, two electrons shared between two AOs

H2 Potential Energy Surface

.

E

. At the dissociation

H + H 0

H H Bond stretching

limit, H2 must separate into two neutral atoms.

H H At the RHF level, the wavefunction, Φ, is 50% ionic and 50% covalent at all χA χ B χA χ A bond lengths.

χB χ B

χB χ A

H2 does not dissociate correctly at the RHF level!! Should be 100% covalent at large internuclear separations.

RHF dissociation problem has several consequences: •

Energies for stretched bonds are too large. Affects transition state structures Ea are overestimated.

•

Equilibrium bond lengths are too short at the RHF level. (Potential well is too steep.) HF method ‘overbinds’ the molecule.

•

Curvature of the PES near equilibrium is too great, vibrational frequencies are too high.

•

The wavefunction contains too much ‘ionic’ character; dipole moments (and also atomic charges) at the RHF level are too large.

On the bright side, SCF procedures recover ~99% of the total electronic energy. But, even for small molecules such as H2, the remaining fraction of the energy - the correlation energy - is ~110 kJ/mol, on the order of a chemical bond.

To overcome the RHF dissociation problem, Use a trial function that is a combination of Φ0 and Φ1 First, write a new wavefunction using the anti-bonding MO.

φ2 = N 2 ( χ A − χ B );

antibonding M O

The form is similar to Φ0, but describes an excited state:

φ 2α (1) φ2 β (1) Φ1 = = φ 2α (1)φ2 β (2) − φ2α (2)φ 2β (1) φ 2α (2) φ2 β (2) Φ1 = φ2 (1)φ 2 (2)[α (1)β (2) − β (1)α (2)]

MO basis

Φ1 = φ2 (1)φ 2 (2) = ( χ A (1) − χ B (1))(χ A (2) − χ B (2)) Φ1 = χ A (1) χ A (2) + χ B (1) χ B (2) − χ A (1) χ B (2) − χ B (1) χ A (2) Ionic terms

Covalent terms

AO basis

Trial function - Linear combination of Φ0 and Φ1; two electron configurations.

Ψ = a0 Φ 0 + a1Φ1 = a0 (φ1φ1) + a1 (φ2φ 2 )

Ψ = (a0 + a1 )[χ A χ A + χ B χ B ]+ (a0 − a1 )[χ A χ B + χ B χ A ] Ionic terms

Covalent terms

Three points: 1. As the bond is displaced from equilibrium, the coefficients (a0, a1) vary until at large separations, a1 = -a0: Ionic terms disappear and the molecule dissociates correctly into two neutral atoms. Ψ = ΨCI, an example of configuration interaction. 2.

The inclusion of anti-bonding character in the wavefunction allows the electrons to be farther apart on average. Electronic motion is correlated.

3.

The electronic energy will be lower (two variational parameters).

Configuration Interaction - Excited Slater Determinants Since the HF method yields the best single determinant wavefunction and provides about 99% of the total electronic energy, it is commonly used as the reference on which subsequent improvements are based. As a starting point, consider as a trial function a linear combination of Slater determinants:

Ψ = a0 Φ HF + ∑ aiΦ i

Multi-determinant wavefunction

i =1

a0 is usually close to 1 (~0.9).

• M basis functions yield M molecular orbitals. • For N electrons, N/2 orbitals are occupied in the RHF wavefunction. • M-N/2 are unoccupied or virtual (anti-bonding) orbitals.

Generate excited Slater determinants by promoting up to N electrons from the N/2 occupied to M-N/2 virtuals: b a

9

a,b,c… =

a

8

virtual MOs

a,b

b a

b a

c

c,d

k i

k,l i

j

j

Ψijkabc

abcd Ψijkl

7 6 5

i,j

i

i

4 3

i,j,k… = occupied MOs 2

j

1

ΨHF Excitation level

Ref.

Ψia Single

Ψijab Double

Ψijab

Triple

Quadruple

…

Represent the space containing all N-fold excitations by Ψ(N). Then the COMPLETE CI wavefunction has the form (2) (1) ΨCI = C0Φ HF + Φ + Φ + Φ

Where

(3)

+ ... + Φ

(N )

Φ HF = Hartree − Fock occ virt

Φ = ∑ ∑ Cia Ψia i a (1)

Φ

occ virt

(2)

= ∑ ∑ Cij Ψij ab

i, j

Φ

Linear combination of Slater determinants with single excitations Doubly excitations

ab

a,b

occ virt

(3)

=∑

∑C

Ψijk

abc ijk

abc

Triples

i, j ,k a,b,c

Φ

(N )

=

occ

virt

∑ ∑

Cijk... Ψijk ... abc...

abc...

N-fold excitation

i, j ,k... a,b,c...

The complete ΨCI expanded in an infinite basis yields the exact solution to the Schrödinger eqn. (Non-relativistic, Born-Oppenheimer approx.)

abc...

The various coefficients, Cijk... , may be obtained in a variety of ways. A straightforward method is to use the Variation Principle.

E CI

ΨCI | H | ΨCI = ΨCI | ΨCI

∂E CI

∂C

abc... ijk...

=0

    H  CK = E K CK

Expectation value of He.

Energy is minimized wrt coeff In a fashion analogous to the HF eqns, the CI Schrodinger equation can be formulated as a matrix eigenvalue problem.

  abc... C C The elements of the vector,  , are the coefficients, ijk... K And the eigenvalue, EK, approximates the energy of the Kth state.

E1 = ECI for the lowest state of a given symmetry and spin. E2 = 1st excited state of the same symmetry and spin, and so on.

Some nomenclature… One-electron basis (one-particle basis) refers to the basis set. This limits the description of the one-electron functions, the Molecular Orbitals. The size of the many-electron basis (N-particle basis) refers to the number of Slater determinants. This limits the description of electron correlation.

In practice, • Complete CI (Full CI) is rarely done even for finite basis sets - too expensive. Computation scales factorially with the number of basis functions (M!). • Full CI within a given one-particle basis is the ‘benchmark’ for that basis since 100% of the correlation energy is recovered. Used to calibrate approximate correlation methods. • CI expansion is truncated at a some excitation level, usually Singles and Doubles (CISD). (1) (2)

ΨCI = C0Φ HF + Φ + Φ

Number of configurations

Example H2O: (19 basis functions)

CISD (~80-90%)

Full CI

Example: Neon Atom

Relative importance Ref. Singles Doubles Triples Quadruples

2 1 4 3 abc...

Weight =

abc... 2 (C ∑ ijk... ) ijk...

for a given excitation level. (Frozen core approx., 5s4p3d basis - 32 functions)

1. 2.

CISD (singles and doubles) is the only generally applicable method. For modest sized molecules and basis sets, ~80-90% of the correlation energy is recovered. CISD recovers less and less correlation energy as the size of the molecule increases.

Multi-configuration Self-consistent Field (MCSCF) 9

Carry out Full CI and orbital optimization within a small active space. Six-electron in six-orbital MCSCF is shown. Written as [6,6]CASSCF.

8 7

Complete Active Space Self-consistent Field (CASSCF)

6

H2O MOs 5 4 3 2 1

ΨHF

Why? 1. To have a better description of the ground or excited state. Some molecules are not welldescribed by a single Slater determinant, e.g. O3. 2. To describe bond breaking/formation; Transition States. 3. Open-shell system, especially low-spin. 4. Low lying energy level(s); mixing with the ground state produces a better description of the electronic state. 5. …

MCSCF Features: 1.

In general, the goal is to provide a better description of the main features of the electronic structure before attempting to recover most of the correlation energy.

2.

Some correlation energy (static correlation energy) is recovered. (So called dynamic correlation energy is obtained through CI and other methods through a large N-particle basis.)

3.

The choice of active space - occupied and virtual orbitals - is not always obvious. (Chemical intuition and experience help.) Convergence may be poor.

4.

CASSCF wavefunctions serve as excellent reference state(s) to recover a larger fraction of the dynamical correlation energy. A CISD calculation from a [n,m]-CASSCF reference is termed Multi-Reference CISD (MRCISD). With a suitable active space, MRCISD approaches Full CI in accuracy for a given basis even though it is not size-extensive or consistent.

Examples of compounds that require MCSCF for a qualitatively correct description. H H C

C

O+

H

H

O

O O-

O

zwitterionic

Singlet state of twisted ethene, biradical.

biradical

H C

N

H C

N

Transition State

O

H

C

N

Mœller-Plesset Perturbation Theory In perturbation theory, the solution to one problem is expressed in terms of another one solved previously. The perturbation should be small in some sense relative to the known problem.

Hˆ = Hˆ 0 + λHˆ ' Hˆ 0 Φ i = E iΦ i, i = 0,1,2,...,∞

Hamiltonian with pert., λ Unperturbed Hamiltonian

Hˆ Ψ = WΨ W = λ 0W0 + λ1W1 + λ 2W2 + ... Ψ = λ 0Ψ0 + λ1Ψ1 + λ 2Ψ2 + ...

As the perturbation is turned on, W (the energy) and Ψ change. Use a Taylor series expansion in λ.

Define Hˆ 0 and Hˆ ' N   ˆ ˆ ˆ ˆ ˆ H0 = ∑ Fi = ∑  hi + ∑ J ij − K ij   i=1  j =1 i =1 N

N

(

N

N

N

)

N

Hˆ '= ∑ ∑ gij − ∑ ∑ gij i =1 j >1

Unperturbed H is the sum over Fock operators ⇒ Moller-Plesset (MP) pert th.

i =1 j =1

Perturbation is a two-electron operator when H0 is the Fock operator.

W0 = sum over M O energies W1 = Φ 0| | Hˆ '| Φ 0 = E(HF) occ vir

W2 = ∑ ∑

ab ab Φ 0| | Hˆ '| Φ ij Φ ij | Hˆ '| Φ 0

E 0 − E ijab

i < j a< b

φφ [ E(MP2) = ∑ ∑ occ vir

i< j a