Ab  Ini'o  Hartree-­‐Fock  Theory  

Basis  Sets  and  LCAO  Wave  Func'ons  

Video  IV.i  

A One-Slide Summary of Quantum Mechanics

Fundamental Postulate:

O ! = a !

operator

wave function

(scalar)observable

What is !? ! is an oracle!

Where does ! come from? ! is refined

Variational Process

H ! = E !

Energy (cannot golower than "true" energy)

Hamiltonian operator(systematically improvable)

electronic road map: systematicallyimprovable by going to higher resolution

convergence of E

truth

What if I can't converge E ? Test your oracle with a question to which youalready know the right answer...

Constructing a 1-Electron Wave FunctionThe units of the wave function are such that its square is electron pervolume. As electrons are quantum particles with non-point distributions,sometimes we say “density” or “probability density” instead of electron pervolume (especially when there is more than one electron, since they areindistinguishable as quantum particles)

For instance, a valid wave function in cartesian coordinates for one electronmight be:

!

" x, y,z;Z( ) =2Z 5 / 2

81 #6$ Z x

2 + y 2 + z 2% & '

( ) * ye$Z x

2 +y2 +z2 / 3

normalizationfactor

radial phasefactor

cartesiandirectionality

(if any)

ensuressquare

integrability

Constructing a 1-Electron Wave FunctionTo permit additional flexibility, we may take our wave function to be a linearcombination of some set of common “basis” functions, e.g., atomic orbitals(LCAO). Thus

!

" r( ) = ai# r( )

i=1

N

$

For example, consider the wavefunction for an electron in a C–H bond.It could be represented by s and pfunctions on the atomic positions, or sfunctions along the bond axis, or anyother fashion convenient. H

C

Constructing a 1-Electron Wave FunctionTo optimize the coefficients in our LCAO expansion, we use the variationalprinciple, which says that

!

E =

ai" i

i

#$

%

& &

'

(

) ) * H a j" j

j

#$

%

& &

'

(

) ) dr

ai" i

i

#$

%

& &

'

(

) ) * a j" j

j

#$

%

& &

'

(

) ) dr

=

aia j

ij

# " i* H" j dr

aia j

ij

# " i* " j dr=

aia j

ij

# ijH

aia j

ij

# ijS

which is minimized by one root E of the “secular equation” (theother roots are excited states)—each value of E that satisfies thesecular equation determines all of the ai values and thus the shapeof the molecular orbital wave function (MO)

H11 – ES11 H12 – ES12 L H1N – ES1N

H21 – ES21 H22 – ES22 L H2N – ES2N

M M O M

HN1 – ESN1 HN2 – ESN2 L HNN – ESNN

= 0

What Are These Integrals H?The electronic Hamiltonian includes kinetic energy, nuclear attraction, and,if there is more than one electron, electron-electron repulsion

The final term is problematic. Solving for all electrons at once is amany-body problem that has not been solved even for classicalparticles. An approximation is to ignore the correlated motion of theelectrons, and treat each electron as independent, but even then, ifeach MO depends on all of the other MOs, how can we determineeven one of them? The Hartree-Fock approach accomplishes thisfor a many-electron wave function expressed as anantisymmetrized product of one-electron MOs (a so-called Slaterdeterminant)

!

Hij = " i #1

2

$2 " j # " i

Zk

rkk

nuclei

% " j + " i

&2

rnn

electrons

% " j

Choose a basis set

Choose a molecular geometry q(0)

Compute and store all overlap, one-electron, and two-electron

integralsGuess initial density matrix P(0)

Construct and solve Hartree- Fock secular equation

Construct density matrix from occupied MOs

Is new density matrix P(n)

sufficiently similar to old

density matrix P(n–1) ?

Optimize molecular geometry?

Does the current geometry satisfy the optimization

criteria?

Output data for optimized geometry

Output data forunoptimized geometry

yes

Replace P(n–1) with P(n)

no

yes no

Choose new geometry according to optimization

algorithm

no

yes

Fµ! = µ –1

2"2 ! – Zk

k

nuclei

# µ1

rk

!

+ P$%$%# µ! $%( ) –

1

2µ$ !%( )&

' ( )

µ! "#( ) = $µ%% 1( )$! 1( )1

r12

$" 2( )$# 2( )dr 1( )dr 2( )

P!" = 2 a!ii

occupied

# a"i

The Hartree-Fock procedure

F11 – ES11 F12 – ES12 L F1N – ES1N

F21 – ES21 F22 – ES22 L F2N – ES2N

M M O M

FN1 – ESN1 FN2 – ESN2 L FNN – ESNN

= 0

One MO per root E

• Slater Type Orbitals (STO)

– Can’t do 2 electron integrals analytically

!

1s" =

#

$

%

& '

(

) *

12

e# +r

( ) ( ) ( ) ( )21

12

221

11 drdrr

!"#µ $$$$% %

What functions to use?

• 1950s– Replace with something similar that is

analytical: a gaussian function

!

1s" =

#

$

%

& '

(

) *

12

e# +r

!

"1s"" =

2#

$

%

& '

(

) *

34

e+#r 2

VS.

Approximating Slater Functions with Gaussians

!

" r( ) = ai# r( )

i=1

N

$

primitive gaussians

STO-3G

Slater function

Contracted basis function

!

"1s"" = ai

i

N

#2$i%

&

' (

)

* +

34

e,$

ir2

Contracted Basis Set

• STO-#G - minimal basis• Pople - optimized a and α values

• What do you about very different bondingsituations?– Have more than one 1s orbital

• Multiple-ζ(zeta) basis set– Multiple functions for the same atomic orbital

H F vs. H H

• Double-ζ – one loose, one tight– Adds flexibility

• Triple-ζ – one loose, one medium, one tight• Only for valence

• Decontraction– Allow ai to vary

• Pople - #-##G– 3-21G!

"1s

H

STO#3G= ai

i

3

$2%i&

'

( )

*

+ ,

34

e#%

ir2

locked in STO-3G

primitive gaussians

primitives in all core functions

valence: 2 tight, 1 loose

gaussian

How many basis functions are for NH3 using 3-21G?

NH3 3-21G

atom # atoms AO degeneracy basis fxns primitives total basis fxns total primitives

N 1 1s(core) 1 1 3 1 3

2s(val) 1 2 2 + 1 = 3 2 3

2p(val) 3 2 2 + 1 = 3 6 9

H 3 1s(val) 1 2 2 + 1 = 3 6 9

total = 15 24

Polarization Functions

• 6-31G**

6 primitives1 core basis functions

2 valence basis functionsone with 3 primitives, the other with 1

1 d functions on all heavy atoms (6-fold deg.)

1 p functions on all H (3-fold deg.)

• For HF, NH3 is planar with infinite basis setof s and p basis functions!!!!!

• Better way to write – 6-31G(3d2f, 2p)• Keep balanced

Valence split polarization

2 d, p

3 2df, 2pd

4 3d2fg, 3p2df

O+

O

H H

= O

Dunning basis set

cc-pVNZ

correlation consistent

polarized

N = D, T, Q, 5, 6

Diffuse functions• “loose” electrons

– anions– excited states– Rydberg states

• Dunning - aug-cc-pVNZ– Augmented

• Pople– 6-31+G - heavy atoms/only with valence– 6-31++G - hydrogens

• Not too useful

How many basis functions in H2POHusing 6-31G(d, p)?