Basics of ab Initioo c a bita o

CHEM 430

Quantum Mechanics

Ab initio

Schrodinger equation

Eigenvalues, eigenfunctions, operators, observables, wavefunctions

Molecular Mechanics

Classical physics

F=ma

Continuum of values

Dirac, early 1900s

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 Function

A valid wave function in cartesian coordinates for one electron might 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

P

x1" x " x

2

y1" y " y

2

z1" z " z

2

#

$

% % %

&

'

( ( (

= )2dx dy dz

z1

z2*y1

y2*x1

x2*

This permits us to compute the

probability of finding the electron

within a particular cartesian volume

element (normalization factors are

determined by requiring that P = 1

when all limits are infinite, i.e.,

integration over all space)

Constructing a 1-Electron Wave Function

To permit additional flexibility, we may take our wave function to be a linear

combination of some set of common “basis” functions, e.g., atomic orbitals

(LCAO). Thus

" r( ) = ai# r( )

i=1

N

$

For example, consider the wave

function for an electron in a C–H bond.

It could be represented by s and p

functions on the atomic positions, or s

functions along the bond axis, or any

other fashion convenient. H

C

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 a

many-body problem that has not been solved even for classical

particles. An approximation is to ignore the correlated motion of the

electrons, and treat each electron as independent, but even then, if

each MO depends on all of the other MOs, how can we determine

even one of them? The Hartree-Fock approach accomplishes this

for a many-electron wave function expressed as an

antisymmetrized product of one-electron MOs (a so-called Slater

determinant)

Hij = " i #1

2

$2 " j # " i

Zk

rkk

nuclei

% " j + " i

&2

rnn

electrons

% " j

5-4 Slater Determinants and the Pauli PrincipleIt was pointed out by Slater [5] that there is a simple way to write wavefunctionsguaranteeing that theywill be antisymmetric for interchange of electronic space and spincoordinates: one writes the wavefunction as a determinant. For the 1s22s configurationof lithium, one would write

ψ =1√6

1s(1)α( 1) 1s(2)α( 2) 1s(3)α( 3)1s(1)β( 1) 1s(2)β( 2) 1s(3)β( 3)2s(1)α( 1) 2s(2)α( 2) 2s(3)α( 3)

(5-38)

Expanding this according to the usual rules governing determinants (seeAppendix 2)gives

ψ =1√6

[1s(1)α( 1)1s(2)β( 2)2s(3)α( 3) + 2s(1)α( 1)1s(2)α( 2)1s(3)β( 3)

+ 1s(1)β( 1)2s(2)α( 2)1s(3)α( 3) − 2s(1)α( 1)1s(2)β( 2)1s(3)α( 3)− 1s(1)β( 1)1s(2)α( 2)2s(3)α( 3) − 1s(1)α( 1)2s(2)α( 2)1s(3)β( 3)] (5-39)

This can be factored and shown to be identical to wavefunction (5-37) of the precedingsection.A simplifying notation in common usage is to delete the α, β symbols of the spin-

orbitals and to let a bar over the space orbital signify β spin, absence of a bar beingunderstood to signify α spin. In this notation, Eq. (5-38) would be written

ψ =1√6

1s(1) 1s(2) 1s(3)1s̄(1) 1s̄(2) 1s̄(3)2s(1) 2s(2) 2s(3)

(5-40)

The general prescription to follow in writing a Slater determinantal wavefunction isvery simple:

1. Choose the configuration to be represented. 1s1s̄2s was used above. (Here we write1s1s̄2s rather than 1s22s to emphasize that the two 1s electrons occupy different spin-orbitals.) For our general example, we will let Ui stand for a general spin-orbitaland take a four-electron example of configuration U1U2U3U4.

2. For n electrons, set up an n × n determinant with (n!)− 1/2 as normalizing factor.Every position in the first row should be occupied by the first spin-orbital of theconfiguration; every position in the second row by the second spin-orbital, etc. Nowput in electron indices so that all positions in column 1 are occupied by electron 1,column 2 by electron 2, etc.

In the case of our four-electron configuration, the recipe gives

ψ =1

√4!

U1(1) U 1(2) U 1(3) U 1(4)U2(1) U 2(2) U 2(3) U 2(4)U3(1) U 3(2) U 3(3) U 3(4)U4(1) U 4(2) U 4(3) U 4(4)

(5-41)

Born-Oppenheimer approximation

Nuclei move much more slowly than electrons (mnuclei ≈ 1800melectron ) Decouple nuclear and electronic motion (find electronic energies for fixed nuclear positions) Nuclei move on a potential energy surface which is a solution to the electronic Schrodinger equation

PES is independent of nuclear mass

Allows us to determine equilibrium and transition state geometries

Nucleus

Electron 16

1.7 Classical Mechanics1.7.1 The Sun–Earth system

The motion of the Earth around the Sun is an example of a two-body system that canbe treated by classical mechanics. The interaction between the two “particles” is thegravitational force.

(1.28)

The dynamical equation is Newton’s second law, which in differential form can bewritten as in eq. (1.29).

(1.29)

The first step is to introduce a centre of mass system, and the internal motion becomesmotion of a “particle” with a reduced mass given by eq. (1.30).

(1.30)

Since the mass of the Sun is 3 × 105 times larger than that of the Earth, the reducedmass is essentially identical to the Earth’s mass (m = 0.999997mEarth). To a very goodapproximation, the system can therefore be described as the Earth moving around theSun, which remains stationary.

The motion of the Earth around the Sun occurs in a plane, and a suitable coordi-nate system is a polar coordinate system (two-dimensional) consisting of r and q.

m =+

=+( )

≅M mM m

mm M

mSun Earth

Sun Earth

Earth

Earth SunEarth1

− =∂∂

∂∂

Vr

rm

t

2

2

V r121 2

12( ) = −C

m mr

grav

θ

rx = rcosθ y = rsinθ

y

x

Figure 1.3 A polar coordinate system

The interaction depends only on the distance r, and the differential equation(Newton’s equation) can be solved analytically.The bound solutions are elliptical orbitswith the Sun (more precisely, the centre of mass) at one of the foci, but for most ofthe planets, the actual orbits are close to circular. Unbound solutions corresponding tohyperbolas also exist, and could for example describe the path of a (non-returning)comet.

Each bound orbit can be classified in terms of the dimensions (largest and smallestdistance to the Sun), with an associated total energy. In classical mechanics, there areno constraints on the energy, and all sizes of orbits are allowed.

Bound and unbound solutions to the classical two-body problem

A Hartree–Fock model for the solar system

Modelling the solar system with actual interactions

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

– Basis set

• set of mathematical functions fromwhich the wavefunction is constructed

– Linear Combination of Atomic Orbitals(LCAO)

Wavefunction

• 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?

Fock Operator

• 2 electron integrals scale as N4

Fµ" = µ #$2

2" # µ

%kr"

k

& + P'( µ" '( #1

2µ' "(

)

* + ,

- . '(

&

P"# = 2 a"ia#ii

occupied MOs

$Density Matrix2 electron integral

Treat electrons as

average field

• 1950s

– Replace with something similar that is

analytical: a gaussian function

1s" =

#

$

%

& '

(

) *

12

e# +r

"1s"" =

2#

$

%

& '

(

) *

34

e+#r 2

VS.

Sir John Pople

British-Americanmathematicianrevolutionizedcomputationalchemistry with avery simple idea!

GAUSSIANS...

Nobel Prize 1998

Properties of Gaussian Functions

Product of Two Gaussians is Another Gaussian

!"$%('%())+,)- · !+$

%('%(-)+,-- = !"!+$%('%())+,)-

%('%(-)+,-- = !"!+$%,--('%())%,)-('%(-)

+,)-,--

Integral of a Gaussian over All of Space is Simple

0 !12

%2$%

('%()-+,- 34 = √2! · |8| · √9

"1s"" = ai

i

N

#2$i%

&

' (

)

* +

34

e,$

ir2

Contracted Basis Set

• STO-#G - minimal basis

• Pople - optimized a and α values

• What to do about very different bonding

situations?

– 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 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 primitives

1 core basis functions

2 valence basis functions

one 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 set

of 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 Sets

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