Lecture 11. Quantum Mechanics. Hartree-Fock Self-Consistent-Field Theory

Outline of today’s lecture

• Postulates in quantum mechanics…• Schrödinger equation…• Simplify Schrödinger equation:

Atomic units, Born-Oppenheimer approximation• Solve Schrödinger equation with approximations:

Variation principle, Slater determinant, Hartree approximation,Hartree-Fock, Self-Consistent-Field, LCAO-MO, Basis set

References

• Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch. 1 & 8

• Essentials of Computational Chemistry. Theories and Models, C. J. Cramer, (2nd Ed. Wiley, 2004) Chapter 4• Molecular Modeling, A. R. Leach (2nd ed. Prentice Hall, 2001) Chapter 2• Introduction to Computational Chemistry, F. Jensen (1999) Chapter 3

• A Brief Review of Elementary Quantum Chemistryhttp://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html

• Molecular Electronic Structure Lecture http://www.chm.bris.ac.uk/pt/harvey/elstruct/introduction.html

• Wikipedia (http://en.wikipedia.org): Search for Schrödinger equation, etc.

Potential Energy Surface & Quantum Mechanics

3N (or 3N-6 or 3N-5) Dimension PES for N-atom system

x

For geometry optimization, evaluate E, E’ (& E’’) at the input structure X(x1,y1,z1,…,xi,yi,zi,…,xN,yN,zN) or {li,θi,i}.

How do we obtain the potential energy E?• MM: Evaluate analytic functions (FF)• QM: Solve Schrödinger equation

li

θi

rij

i

QuantumModeling

AtomisticModeling

ContinuumModeling

ChargeForce Field

Geometry

1m 1cm0.1 nm

1 nm

Length

The Schrödinger Equation

The ultimate goal of most quantum chemistry approach is the solution of the time-independent Schrödinger equation.

Hamiltonian operator wavefunction(solving a partial differential equation)

(1-dim) ?

Postulate #1 of Quantum Mechanics

•The state of a quantum mechanical system is completely specified by the wavefunction or state function that depends on the coordinates of the particle(s) and on time.

•The probability to find the particle in the volume element located at r at time t is given by . (Born interpretation)

•The wavefunction must be single-valued, continuous, finite, and normalized (the probability of find it somewhere is 1).

= <|>

dtrtr ),(),(

),( trΨ

drdtd

1),(2 trd

Probabilitydensity

Born Interpretation of the Wavefunction: Probability Density

B = 0B = 0

A = 0A = 0

A = BA = B

ikxAe

ikxBe

kxAcos2

22A

22B

kxA 22cos4

nodes

Probability Density: Examples

Postulate #2 of Quantum Mechanics

• Once is known, all properties of the system can be obtained

by applying the corresponding operators to the wavefunction.

• Observed in measurements are only the eigenvalues a which satisfy

the eigenvalue equation

with (Hamiltonian operator)

(e.g. with )

),( trΨ

Schrödinger equation: Hamiltonian operator energy

(Operator)(function) = (constant number)(the same function)

(Operator corresponding to observable) = (value of observable)

eigenvalue eigenfunction

Physical Observables & Their Corresponding Operators

with (Hamiltonian operator)

(e.g. with )

Observables, Operators & Solving Eigenvalue Equations:an Example

ikxAe

dx

d

ipx

ˆ

xpdx

d

i

khkhAeAedx

d

iikxikx

khpx constantnumber

the same function

The Uncertainty Principle

When momentum is known precisely, the position cannot be predicted precisely, and vice versa.

When the position is known precisely,

Location becomes precise at the expense

of uncertainty in the momentum

ikxAe 22Akhpx

Postulate #3 of Quantum Mechanics

• Although measurements must always yield an eigenvalue,

the state does not have to be an eigenstate.

• An arbitrary state can be expanded in the complete set of

eigenvectors ( as where n .

• We know that the measurement will yield one of the values ai, but

we don't know which one. However,

we do know the probability that eigenvalue ai will occur ( ).

• For a system in a state described by a normalized wavefunction ,

the average value of the observable corresponding to is given by

= <|A|> dAA

The Schrödinger Equation for Atoms & Molecules

Atomic Units (a.u.)

• Simplifies the Schrödinger equation (drops all the constants)

(energy) 1 a.u. = 1 hartree = 27.211 eV = 627.51 kcal/mol,(length) 1 a.u. = 1 bohr = 0.52918 Å,(mass) 1 a.u. = electron rest mass,(charge) 1 a.u. = elementary charge, etc.

(before) (after)

Born-Oppenheimer Approximation

• Simplifies further the Schrödinger equation (separation of variables)

• Nuclei are much heavier and slower than electrons.• Electrons can be treated as moving in the field of fixed nuclei.

• A full Schrödinger equation can be separated into two:– Motion of electron around the nucleus– Atom as a whole through the space

• Focus on the electronic Schrödinger equation

E =

(before)

(after)

(electronic)

(nuclear)

Born-Oppenheimer Approximation

Electronic Schrödinger Equation in Atomic Unit

Variational Principle

1. Nuclei positions/charges & number of electrons in the molecule2. Set up the Hamiltonian operator3. Solve the Schrödinger equation for wavefunction , but how?4. Once is known, properties are obtained by applying operators

• No exact solution of the Schrödinger eq for atoms/molecules (>H)

• Any guessed trial is an upper bound to the true ground state E.

• Minimize the functional E[] by searching through all acceptableN-electron wavefunctions

==

Hartree Approximation (1928)Single-Particle Approach

• Impossible to search through all acceptable N-electron wavefunctions.

• Let’s define a suitable subset.• N-electron wavefunction

is approximated by a product of N one-electron wavefunctions. (Hartree product)

Nobel lecture (Walter Kohn; 1998) Electronic structure of matter

Antisymmetry of Electrons and Pauli’s Exclusion Principle

• Electrons are indistinguishable. Probability doesn’t change.

• Electrons are fermion (spin ½). antisymmetric wavefunction

• No two electrons can occupy the same state (space & spin).

Slater “determinants”

• A determinant changes sign when two rows (or columns) are exchanged.

Exchanging two electrons leads to a change in sign of the wavefunction. • A determinant with two identical rows (or columns) is equal to zero.

No two electrons can occupy the same state. “Pauli’s exclusion principle”

“antisymmetric”

= 0 = 0

N-Electron Wavefunction: Slater Determinant

• N-electron wavefunction aprroximated by a product of N one-electron

wavefunctions (hartree product).

• It should be antisymmetrized ( ).Hartree product is not antisymmetric!

Hartree-Fock (HF) Approximation

• Restrict the search for the minimum E[] to a subset of , which is all antisymmetric products of N spin orbitals (Slater determinant) • Use the variational principle to find the best Slater determinant (which yields the lowest energy) by varying spin orbitals

(orthonormal)= ij

Hartree-Fock (HF) Energy

Hartree-Fock (HF) Energy: Evaluation

Molecular Orbitals as linear combinations of Atomic Orbitals (LCAO-MO)

(spin orbital = spatial orbital * spin)

where

Slater determinant

finite “basis set”

• No-electron contribution (nucleus-nucleus repulsion: just a constant)

• One-electron operator h (depends only on the coordinates of one electron)

• Two-electron contribution (depends on the coordinates of two electrons)

Hartree-Fock (HF) Equation: Evaluation

where

1. Potential energy due to nuclear-nuclear Coulombic repulsion (VNN)

2. Electronic kinetic energy (Te)

3. Potential energy due to nuclear-electronic Coulombic attraction (VNe)

*In some textbooks ESD doesn’t include VNN, which will be added later (Vtot = ESD + VNN).

3. Potential energy due to two-electron interactions (Vee)

• Coulomb integral Jij (local)

Coulombic repulsion between electron 1 in orbital i and electron 2 in orbital j

• Exchange integral Kij (non-local) only for electrons of like spins

No immediate classical interpretation; entirely due to antisymmetry of fermions

> 0, i.e., a destabilization

Hartree-Fock (HF) Energy: Integrals

Self-Interaction

• Coulomb term J when i = j (Coulomb interaction with oneself)

• Beautifully cancelled by exchange term K in HF scheme = 0

0

Constrained Minimization of EHF[SD]

Hartree-Fock (HF) Equation

Two-electron repulsion cannot be separated exactly into one-electron terms. By imposing the separability, the Molecular Orbital Approximation inevitably involves an incorrect treatment of the way in which the electrons interact with each other.

• Fock operator: “effective” one-electron operator

and

• two-electron repulsion operator (1/rij) replaced by one-electron operator VHF(i)

by taking it into account in “average” way

Self-Consistent Field (HF-SCF) Method

• Fock operator depends on the solution.

• HF is not a regular eigenvalue problem that can be solved in a closed form.

1. Start with a guessed set of orbitals;2. Solve HF equation;3. Use the resulting new set of orbitals

in the next iteration; and so on;4. Until the input and output orbitals

differ by less than a preset threshold(i.e. converged).

Koopman’s Theorem

• As well as the total energy, one also obtains a set of orbital energies.

• Remove an electron from occupied orbital a.

Orbital energy = Approximate ionization energy

Koopman’s Theorem: Examples

Restricted vs Unrestricted HF

Accuracy of Molecular Orbital (MO) Theory

Electron Correlation

• A single Slater determinant never corresponds to the exact wavefunction.

EHF > E0 (the exact ground state energy)

• Correlation energy: a measure of error introduced through the HF scheme

EC = E0 EHF (< 0)

– Dynamical correlation– Non-dynamical (static) correlation

• Post-Hartree-Fock method– Møller-Plesset perturbation: MP2, MP4– Configuration interaction: CISD, QCISD, CCSD, QCISD(T)

Summary