The Hartree-Fock Approximation

Shyue Ping Ong

Stationary Schrödinger Equation for a System of Atoms

where

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Eψ = Hψ

H = −h 2

2me

∇i2

i∑ −

h 2

2mk

∇k2 −

e2Zk

rikk∑

i∑ +

e2

rijj∑

i∑

k∑ +

ZkZle2

rkll∑

k∑

KE of electrons

KE of nuclei

Coulumbic attraction between nuclei and electrons

Coulombic repulsion between electrons

Coulombic repulsion between nuclei

Stationary Schrödinger Equation in Atomic Units

To simplify the equations a little, let us from henceforth work with atomic units

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Dimension Unit Name Unit Symbol Mass Electron rest mass me

Charge Elementary Charge e Action Reduced Planck’s constant ħ Electric constant Coulomb force constant ke

H = −12∇i2

i∑ −

12mk

∇k2 −

Zk

rikk∑

i∑ +

1rijj

∑i∑

k∑ +

ZkZlrkll

∑k∑

The Variational Principle

We can judge the quality of the wave functions by the energy – the lower the energy, the better. We may also use any arbitrary basis set to expand the guess wave function.

How do we actually use this?

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φHφ dr∫φ 2 dr∫

≥ E0

Linear combination of atomic orbitals (LCAO)

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http://www.orbitals.com/

Solving the one-electron molecular system with the LCAO basis set approach

In general, we may express our trial wave functions as a series of mathematical functions, known as a basis set. For a single nucleus, the eigenfunctions are effectively the hydrogenic atomic orbitals. We may use these atomic orbitals as a basis set for our molecular orbitals. This is known as the linear combination of atomic orbitals (LCAO) approach.

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φ = aiϕii=1

N

∑

The Secular Equation

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E =aiϕi

i=1

N

∑"

#$

%

&'H aiϕi

i=1

N

∑"

#$

%

&'dr∫

aiϕii=1

N

∑"

#$

%

&'

2

dr∫

=

aiaj ϕiHϕ j dr∫ij∑

aiaj ϕiϕ j dr∫ij∑

=

aiajHijij∑

aiajSijij∑

Resonance integral

Overlap integral

The Secular Equation, contd

To minimize the energy,

Which gives

Or in matrix form

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∂E∂ak

= 0, ∀k

ai (Hki −ESki )i=1

N

∑ = 0, ∀k

H11 −ES11 H12 −ES12 ! H1N −ES1NH21 −ES21 H22 −ES22 ! H2N −ES2N" " # "

HN1 −ESN1 HN 2 −ESN 2 ! HNN −ESNN

"

#

$$$$$

%

&

'''''

a1a2"aN

"

#

$$$$$

%

&

'''''

= 0

The Secular Equation, contd

Solutions exist only if Procedure:

i.  Select a set of N basis functions. ii.  Determine all N2 values of Hij and Sij. iii.  Form the secular determinant and determine the N roots Ej. iv.  For each Ej, solve for coefficients ai.

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H11 −ES11 H12 −ES12 ! H1N −ES1NH21 −ES21 H22 −ES22 ! H2N −ES2N" " # "

HN1 −ESN1 HN 2 −ESN 2 ! HNN −ESNN

= 0

Hückel Theory

Basis set formed from parallel C 2p orbitals Overlap matrix is given by Hii = Ionization potential of methyl radical Hij for nearest neighbors obtained from exp and 0 elsewhere

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Sij = δij

The Born-Oppenheimer Approximation

Heavier nuclei moves much more slowly than electrons => Electronic relaxation is “instantaneous” with respect to nuclear motion

Electronic Schrödinger Equation

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(Hel +VN )ψel (qi;qk ) = Eelψel (qi;qk )

Electronic energy Constant for a set of nuclear coordinates

Stationary Electronic Schrödinger Equation

where

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Eelψel = Helψel

Hel = −12∇i2

i∑ −

Zk

rikk∑

i∑ +

1rijj

∑i∑

KE and nuclear attraction terms are separable

H = hii∑ where hi = −

12∇i −

Zk

rikk∑

Hartree-Product Wave Functions

Eigen functions of the one-electron Hamiltonian is given by

Because the Hamiltonian is separable,

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hiψi = εiψi

ψHP = ψii∏

HψHP = hii∑ ψk

k∏

= εii∑#

$%

&

'(ψHP

The effective potential approach

To include electron-electron repulsion, we use a mean field approach, i.e., each electron sees an “effective” potential from the other electrons

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hi = −12∇i −

Zk

rikk∑ +Vi, j

where

Vi, j =ρ j

rij∫

j≠i∑ dr

Hartree’s Self-Consistent Field (SCF) Approach

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Guess MOs

Construct one-

electron operations

hi

Solve for new ψ

hiψi = εiψi

Iterate until energy eigenvalues converge to a desired level of accuracy

E = εii∑ −

12

ψi2ψ j

2

rijdri drj∫∫

What’s the purpose of this term?

What about the Pauli Exclusion Principle?

Two identical fermions (spin ½ particles) cannot occupy the same quantum state simultaneously

è Wave function has to be anti-symmetric

For two electron system, we have

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ψSD =12ψa (1)α(1)ψb(2)α(2)−ψa (2)α(2)ψb(1)α(1)[ ]

=12

ψa (1)α(1) ψb(1)α(1)ψa (2)α(2) ψb(2)α(2)

where α is the electron spin eigenfunction

Slater determinant

For many electrons…

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ψSD =1N!

χ1(1) χ2 (1) ! χN (1)χ1(2) χ2 (2) ! χN (2)! ! " !

χ1(N ) χ2 (N ) ! χN (N )

where χ k are the spin orbitals

The Hartree-Fock (HF) Self-Consistent Field (SCF) Method

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fi = −12∇i2 −

Zk

rik+Vi

HF{ j}k

nuclei

∑

F11 −ES11 F12 −ES12 ! F1N −ES1NF21 −ES21 F22 −ES22 ! F21 −ES2N" " # "

FN1 −ESN1 FN 2 −ESN 2 ! FNN −ESNN

= 0

HF Secular Equation

Fµυ = µ |− 12∇i2 |υ − Zk µ | 1

rk|υ + Pλσ

λσ

∑ (µυ | λσ )− 12(µλ |υσ )

$

%&'

()k

nuclei

∑

Weighting of four-index integrals by density matrix, P

Flowchart of HF SCF Procedure

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Limitations of HF

Fock operators are one-electron => All electron correlation, other than exchange, is ignored

Four-index integrals leads to N4 scaling with respect to basis set size

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Ecorr = Eexact −EHF

Practical Aspects of HF Calculations

Basis Sets

Effective Core

Potentials

Open-shell vs Closed-

shell Accuracy

Performance

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Basis Set

Set of mathematical functions used to construct the wave function.

In theory, HF limit is achieved by an infinite basis set.

In practice, use finite basis sets that can approach HF limit as efficiently as possible

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Contracted Gaussian Functions

Slater-type orbitals (STO) with radial decay cannot be analytically integrated

-> Use linear combination of Gaussian-type orbitals (GTOs) with radial decay to approximate STOs

STO-3G •  STO approximated by 3 GTOs •  Known as single-ζ or minimal basis set.

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e−r2

e−r

Multiple-ζ and Split-Valence

Multiple-ζ •  Adding more basis functions per atomic orbital •  Examples: cc-pCVDZ, cc-pCVTZ (correlation-consistent polarized

Core and Valence (Double/Triple/etc.) Zeta)

Split-valence or Valence-Multiple-ζ •  Still represent core orbitals with single, contracted basis functions •  Valence orbitals are split into many functions (Why?) •  Examples: 3-21G, 6-31G, 6-311G

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# of primitives in core

# of primitives in valence

Polarization and Diffuse Functions

Polarization functions •  Description of MOs require more flexibility than provided by AOs,

e.g., NH3 is predicted to be planar if using just s and p functions •  Additional basis functions of one quantum number of higher

angular momentum than valence, e.g., first row -> d orbitals •  Notation: 6-31G* [old] or 6-31G(d) [new], 6-31(2d,p) [2d functions

for heavy atoms, additional p for H]

Diffuse functions

•  Highest energy MOs of anions, highly excited states tend to be more diffuse

•  Augment standard basis sets with diffuse functions •  Notation: 6-31+G, 6-311++G(3df, 2pd), aug-cc-pCVDZ

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Effective Core Potentials

Heavy atoms have many electrons •  Intractable to model all of them, even with a minimal basis set •  However, most of the electrons are in the core

Solution: Replace core electrons with analytical functions (effective core potentials or ECPs) that represent combined nuclear-electronic core to the remaining electrons Key selection decision: How many electrons to include in the core?

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Open-shell vs closed-shell

Restricted HF (RHF) •  Closed-shell systems, i.e., no unpaired electrons

Restricted open-shell HF (ROHF)

•  Use RHF formalism, but with density matrix for singly occupied orbitals not multiplied by a factor of 2.

•  Wave functions are eigenfunctions of S2

•  But fails to account for spin polarization in doubly occupied orbitals

Unrestricted HF (UHF)

•  Includes spin polarization •  Wave functions are not eigenfunctions of S2, i.e., spin contamination

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Accuracy

Energetics •  In general, extremely poor; correlation is extremely important in

chemical bonding! •  Protonation energies are typically ok (no electrons in H+) •  Koopman’s Theorem: First IE is equal to the negative of the

orbital energy of the HOMO

Geometry

•  Typically relatively good ground state structures with basis sets of modest size

•  But transition states (with partial bonding) can be problematic, as well as some pathological systems

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Performance

Formal N4 scaling But in reality, speedups can be achieved through:

•  Symmetry •  Estimating upper bounds to four-index integrals •  Fast multipole and linear exchange integral computations

For practical geometry optimizations, frequently helps to first compute geometry with a smaller basis set to provide a better initial geometry and a guess for the Hessian matrix.

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