Transcript Introduction to Electronic Structure Theory

Introduction to Computational Chemistry
NSF Computational Nanotechnology and Molecular Engineering
Pan-American Advanced Studies Institutes (PASI) Workshop
January 5-16, 2004
California Institute of Technology, Pasadena, CA
Andrew S. Ichimura
For the Beginner…
There are three main problems:
1. Deciphering the language.
2. Technical implementation.
3. Quality assessment.
Focus on…
Calculating molecular structures and relative
energies.
1. Hartree-Fock (Self-Consistent Field)
2. Electron Correlation
3. Basis sets and performance
Molecular
properties
Transition States
Reaction coords.
Ab initio electronic structure theory
Hartree-Fock (HF)
Electron Correlation (MP2, CI, CC, etc.)
Geometry
prediction
Benchmarks for
parameterization
Spectroscopic
observables
Prodding
Experimentalists
Goal: Insight into chemical phenomena.
Setting up the problem…
What is a molecule?
A molecule is “composed” of atoms, or, more generally as a collection of charged
particles, positive nuclei and negative electrons.
The interaction between charged particles is described by;
qiq j
qiq j
Vij  V (rij ) 

4 0 rij
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, Y.
Y
ˆ
HY  i
t
Time-dependent Schrödinger Equation
i 
Hˆ


1;  h 2

Hamiltonian Operator


Time-Independent Schrödinger Equation
Hˆ (r,t)  Hˆ (r)
Y(r,t)  Y(r)eiEt /
Hˆ (r)Y(r)  EY(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 particle-wave duality of electrons.



Hamiltonian for a system with N-particles
Hˆ  Tˆ  Vˆ
Sum of kinetic (T) and potential (V) energy
2
2 
  2


Tˆ  Tˆi  
  
 2  2  2 
y i zi 
i1
i1 2mi
i1 2mi x i
N
N
N
2
2
i
  2
 2  2 
   2  2  2 
x i y i zi 
2
i
N
N
N
2
Kinetic energy
Laplacian operator
N
qiq j
ˆ
V  Vij  
rij
i1 j1
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:

ˆ  Tˆ  Tˆ  Vˆ  Vˆ  Vˆ
H
n
e
ne
ee
nn
Tˆn
Tˆe
Vˆne
Vˆee
Vˆnn
•
Kinetic energy of nuclei, n
Kinetic energy of electrons, e
Electron-nuclear attraction
Electron-electron repulsion
Internuclear repulsion
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
0
H
•
.
H + 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 (Z<36), but not for the 4th period or higher.
• By neglecting relativistic effects, electron spin must be introduced in an ad
hoc fashion. Spin-dependent terms, e.g., spin-orbit or spin-spin coupling may
be calculated as corrections after the electronic Schrödinger equation has
been solved.
The electronic Hamiltonian becomes,
ˆ  Tˆ  Vˆ  Vˆ  Vˆ
H
e
ne 
ee
nn
B.O. approx.; fixed nuclear coord.
Self-consistent Field (SCF) Theory
GOAL: Solve the electronic Schrödinger equation, HeY=EY.
PROBLEM: Exact solutions can only be found for one-electron systems,
e.g., H2+.
SOLUTION: Use the variational principle to generate approximate
solutions.
Variational principle - If an approximate wavefunction is used in
HeY=EY, then the energy must be greater than or equal to the exact
energy. The equality holds when Y is the exact wavefunction.
In practice: Generate the “best” trial function that has a number of
adjustable parameters. The energy is minimized as a function of these
parameters.
SCF cont.
The energy is calculated as an expectation value of the Hamiltonian operator:
E
 ˆ
Y
 H eYd

Y
 Yd
Introduce “bra-ket” notation,
 ˆ
Y
 He Yd  Y | Hˆ e | Y

Y
 Yd  Y | Y
bra
n
ket
m
complex conjugate , left
right
Combined bracket denotes integration over all coordinates.
E
Y | Hˆ e | Y
Y |Y
If the wavefunctions are orthogonal and normalized (orthonormal),
ij  1
Yi | Yj   ij
(Kroenecker delta)
 0
ij
Then,
E  Y | Hˆ e | Y
SCF cont.
Antisymmetric wavefunctions can be written as
Slater determinants.
Since electrons are fermions, S=1/2, the total electronic wavefunction must be
antisymmetric (change sign) with respect to the interchange of any two electron
coordinates. (Pauli principle - no two electrons can have the same set of quantum
numbers.)
Consider a two electron system, e.g. He or H2. A suitable antisymmetric
wavefunction to describe the ground state is:
1,2  1(1) 2(2)  1 (2) 2 (1)
(He: 1 =2 = 1s)
(H2: 1 = 2 = bonding MO)
Each electron resides in a spin-orbital, a product of spatial and spin functions.
(Spin funct ions are ort honormal:
 |  =  |  =1;  |    |   0)
Interchange the coordinates of the two electrons,
2,1  1(2) 2(1)  1 (1) 2 (2)
2,1   1,2
SCF cont.
A more general way to represent antisymmetric electronic wavefunctions is in the
form of a determinant. For the two-electron case,
1 (1)  2(1)
1,2 
 1(1) 2(2)  1 (2) 2 (1)
1 (2)  2(2)
For an N-electron N-spinorbital wavefunction,
 SD 
1 1
1 2
 2 (1)
 2 (2)
 N (1)
 N (2)
1 N   2 (N)
 N (N)
,
 i |  j  ij
A Slater Determinant (SD) satisfies the antisymmetry requirement.
Columns are one-electron wavefunctions, molecular orbitals.
Rows contain the electron coordinates.
One more approximation: The trial wavefunction will consist of a single SD.
Now the variational principle is used to derive the Hartree-Fock equations...
Hartree-Fock Equations
(1) Reformulate the Slater Determinant as,
  Aˆ1(1)2(2) N (N) Aˆ 
 is the diagonal product
ˆ t he ant isymmetrizer
A
N 1
1
1 
p ˆ
ˆ
A
(1) P 
1  Pˆij   Pˆij k 

N! p 0
N! 
ij k
 ij




Pˆ is the permut at ion operator.
Pˆij permut es t wo elect ron coordinates.
(2)
ˆ  Tˆ  Vˆ  Vˆ  Vˆ
H
e
e
ne 
ee
nn
N
1
Tˆe   2i
i 2
N
Vˆne   
i
a
Za
Ra  ri

One electron
terms
N
N
Vˆee   
i
j i
1
ri  rj
ZZ
Vˆnn    a b
a b a Ra  Rb
Depends on
two electrons
1 2
Za
hˆi    i  
2
a Ra  ri
gˆij 
One-electron operator - describes electron
i, moving in the field of the nuclei.
1
Two-electron operator - interelectron
repulsion.
ri  r j
N
N
N
i 1
i
j i
Hˆ e   hˆi    gˆij  Vˆnn
Hamiltonian
(3) Calculation of the energy.
E e   | Hˆ e | 
N 1
E e  Aˆ  | Hˆ e | Aˆ    (1) p  | Hˆ e | Pˆ 
Expectation value over
Slater Determinant
p 0
Examine specific integrals:
 | Vˆnn |   Vnn
Nuclear repulsion does not depend
on electron coordinates.
For coordinate 1,
 | hˆ1 |   1 (1) 2 (2)
 N (N)| hˆ1 | 1 (1) 2 (2)  N (N)
 1 (1) | hˆ1 | 1 (1)  2 (2) |  2 (2)
N (N ) | N (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.
 | gˆ12 |   1 (1) 2 (2)  N (N)| gˆ12 | 1(1) 2 (2)  N (N )
 1 (1)2 (2) | gˆ12 | 1 (1)2 (2)  3 (3) |  3 (3)
N (N ) |  N (N )
= 1 (1)2 (2) | gˆ12 | 1 (1)2 (2)  J12
Coulomb integral, J12: represents the classical repulsion
between two charge distributions 12(1) and 22(2).
 | gˆ12 | Pˆ12   1 (1) 2 (2)  N (N)| gˆ12 |  2 (1)1 (2)  N (N)
 1 (1)2 (2) | gˆ12 | 2 (1)1 (2)  3 (3) |  3 (3)
N (N ) |  N (N )
= 1 (1)2 (2) | gˆ12 | 2 (1)1 (2)  K12
Exchange integral, K12: no classical analogue. Responsible for
chemical bonds.
The expression for the energy can now be written as:
N
1 N N
E e   hi    (J ij  Kij )  Vnn
2 i j
i 1
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
 i | hˆi |  i     j | Jˆi |  j   j | Kˆ i |  j
2 i j
 V
nn
Jˆi |  j (2)   i (1) | gˆ12 |  i (1)  j (2)
Kˆ i |  j (2)   i (1) | gˆ12 |  j (1)  i (2)
The objective now is to find the best orbitals (i, MOs) that minimize the
energy (or at least remain stationary with respect to further changes in i),
while maintaining orthonormality of i.
• 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 , )  f (x1, x 2 , x N )  g(x1 ,x 2 , x N )
function.
L
L
OptimizeL such that
 0,
 0 Constrained optimization of L.
xi
 i
• In terms of molecular orbitals, the Langrange function is:

N
L  E   ij  i |  j   ij

ij
N

L  E   ij  i |  j  i |  j

0
ij
Change in L with respect to small
changes in i should be zero.
• Change in the energy with respect changes in i.
N
E  
i 1

 
N
 i | hˆi |  i   i | hˆi | i   i | Jˆ j  Kˆ j |  i  i | Jˆ j  Kˆ j |  i
ij

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.

E    i | Fˆi | i   i | Fˆi | i
i 1
N
L  
i1



Change in energy in terms
of the Fock operator.
N

 i | Fˆi |  i   i | Fˆi |  i   ij  i |  j   i |  j
 0
ij
According to the variational principle, the best orbitals, i, will make L=0.
After some algebra, the final expression becomes:
N
Fˆii   ij j
j
Hartree-Fock Equations
After a unitary transformation, ij0 and iii.
Fˆii '  ii '
i   i '| Fˆi |  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.
Koopman’s Theorem
The ionization energy is well approximated by the orbital energy, i.
* Calculated according to Koopman’s theorem.
Basis Set Approximation
•
•
For atoms and diatomic molecules, numerical HF methods are available.
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
i   ci 

M
M
Fˆi  c i    i  ci 

LCAO - MO representation
Coefficients are variational parameters
HF equations in the AO basis

FC  SC
F   | Fˆ |  
S    |  
Matrix representation of HF eqns.
Roothaan-Hall equations (closed shell)
F - element of the Fock matrix
S - 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
E
i 1


N N
1
 i | hˆi |  i     i j | gˆ |  i j   i j | gˆ |  j i  Vnn
2 i j
Total Energy in AO basis
N
M
E    c ic i
i 1 
N

M
1
 | hˆi |     cic j c icj   | gˆ |       | gˆ |  
2 ij 
One-electron integrals, M2
D 

j
c j cj ;
D 
nn
Two-electron integrals, M4
Computed at the start; do not change
Products of AO coeff form Density Matrix, D
oc c.MO
 V
oc c.MO

i
c ic i
General SCF Procedure
Obtain initial guess
for coeff., ci,form
the initial D
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  and  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)
 and  spinorbitals have different spatial parts. 
5
4
Energy
3
2
1
RHF
Singlet
ROHF
Doublet
UHF
Doublet


Spinorbitals
is(n)
Comparison of RHF and UHF
R(O)HF
•  and  spins have same spatial
part
UHF
•  and  spins have different
spatial parts
• Wavefunction, , is an
eigenfunction of S2 operator.
• Wavefunction is not an
eigenfunction of S2.  may be
contaminated with states of
higher multiplicity (2S+1).
• For open-shell systems, the
unpaired electron () interacts
differently with  and  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.
• EUHF ≤ ER(O)HF
• Yields qualitatively correct
spin densities.
• Starting point for more
advanced calculations that
include electron correlation.
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!!!