Starting to talk about
electronic structure methods
CHEM 430
Spring 20161
Computational chemistry
• Study molecular structures, energetic properties, and molecular
interactions which are difficult to obtain in a lab
– Examples include dangerous molecules, transition states
• Eliminate health and safety issues
• Compare to and aid experimental work
2
Method selection
100 fs, 10
atoms:
photochemistry
Correlated Method
(MP2, CCSD)
Molecular detail
Atomistic MM
10 ms, thousands
of atoms:
protein folding,
drug binding
Density
Functional
Theory
10 ps, 100
atoms: chemical
reactions
Coarse-grain
1 ms+, 1 million
atoms: dynamics of
large proteins, cell
membranes, viruses
Computational cost
3
A few limitations
• Computational cost
– Computer processing unit (CPU) time, memory,
disk space
• Molecular size
– How many heavy (non-hydrogen) atoms?
• QMtreat smaller systems (dozens of
atoms…depending on the method…more later)
• MMLarger systems (e.g., proteins)
4
Distinctions
Molecular mechanics
• Classical physics
• F=ma
• Continuum of values
Quantum Mechanics
• Ab initio
• Schrodinger equation
• Eigenvalues,
eigenfunctions,
operators, observables,
wavefunctions
Dirac, early 1900s
5
Quantum mechanics
Schrodinger equation
Ĥ  E
• Ĥ is the Hamiltonian operator
• Ψ is the wavefunction
• E is the energy
Chemistry
Physics
QM
Mathematics
6
Ĥ  E
Eigen…
Qopψi= qiψi
• Eigenvalue
– Correspond to an eigenfunction (and eigenstate)
– Each eigenfunction has a set of eigenvalues that are
possible from making a measurement on a system
– Eigenwert (Characteristic value)
• Eigenfunction
– Eigenfunktion (Characteristic function)
8
Ĥ  E
Qopψi= qiψi
• Operators (Qop)
– Transform one function to another
– Applied to wavefunctions to determine properties
– Operator for the energy is the Hamiltonian
– Operation of the wavefunction is the Schrodinger equation
– There is an operator for every physical observable
• Observables
– Each observable corresponds to a linear operator
– E.g., position, momentum, energy
9
Ĥ  E
Wavefunctions
• Probability map…where spatial amplitude (dependent on
x,y,z) is large, good chance you will find your system there
• Describe the state of a system
• Contain the measureable information about the system
• Operator on the wavefunction

2
• Must be
 ( x, y, z, t ) dxdydz  1

– 1.) Normalizable
– 2.) Continuous (and its derivative)
– 3.) Single-valued

10
Ĥ  E
Radial and angular
(r, ,  )  R(r )Y ( ,  )
Y ( ,  )  ( )( )
• Solve Schrodinger equation via separation of variables
• r is distance from center of nucleus, θ is the angle to the
positive z-axis, and φ is the angle to the positive x-axis in the
xy-plane
• Each set of quantum numbers (n,l,ml) describes a different
wave function
• Radial depends on n and l
• Angular depends on l and ml
11
Time-dependent Schrodinger equation
  

 V ( x)  i
2
2m x
t
2
2
 

 H
i t
• Wavefunction evolves with time and behaves like a wave
• Newton’s equation (F=ma) gives us position of a particle as a
function of time
• Schrodinger equation gives us Ψ(x,t) where square of the
wavefunction tells us the probability of finding the particle in
space as a function of time
12
Removing the time dependency
 

 H
i t
( x, y, z, t )   ( x, y, z) (t )
  d 
 
   ( H )
 i dt 
1  d 1

 ( H )
 i dt 
1.
2.
3.
4.
(0)
(1)
(2)
(3)
Consider a form of Ψ such that time and position variables can be
separated
Put right-hand side of (1) into (0)
Rearrange (2)
LHS of (3) depends on t and RHS depends on x,y,z (therefore, LHS and
RHS equal the same constant)
13
Removing the time dependency
 d

 E
i dt
1  d 1

 ( H )
 i dt 
 (t )  e iEt / 
( x, y, z, t )   ( x, y, z )eiEt / 
•
•
C
1

( H )
C  ( H )
H  E
RHS is in form of operator equation (H is
Hamiltonian so C is energy, E)
LHS must equal E…1st order differential equation
14
Time-independent
Schrodinger equation
Ĥ  E
Hˆ  Tˆe  Tˆn  Vˆne  Vˆnn  Vˆee
nuclei 1
electrons nuclei e 2 Z
nucleie 2 Z Z
electrons e 2
1
A
A B
Hˆ   
i2    2A   
 
 
i
A 2
i
A
A B
i j
2
riA
rAB
rij
electrons
• To find what the wavefunction is for a certain energy means we
need to remove the time dependence
We can make approximations
to simplify this even more!
15
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
Adiabatic approximation
• The form of the total
wavefunction is restricted
to one electronic surface
• If the system starts in one
eigenstate of the
Hamiltonian, it will
remain in that eigenstate
from initial to final (of an
adiabatic process)
• Time-derivative of the
Hamiltonian is small
This and the BO approximation break down when ≥ 2
solutions to the Schrodinger equation are close energetically
17
Schrödinger Equation
Ĥ  E
Hˆ  Tˆe  Tˆn  Vˆne  Vˆnn  Vˆee
Constant
0
nuclei 1
electrons nuclei e 2 Z
nucleie 2 Z Z
electrons e 2
1
2
2
A
A B
Hˆelec  
i    A   
 
 
i
A 2
i
A
A B
i j
2
riA
rAB
rij
electrons
• Assume: Born-Oppenheimer approximation
Nucleus
Electron
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Scaling
Method and basis set
Number of basis functions
CBS
“Exact”
solution to
HΨ=E Ψ
6Z
5Z
QZ
TZ
DZ
Hartree-Fock DFT MP2
CCSD
CCSD(T)
Full CI
Increasing Cost
• Scaling
• Memory, computer processing time
19
Desirable qualities in a method
1.
2.
3.
4.
Size extensive
Size consistent
Variational
Accurate
20
Size extensive
Energy
• Correct (linear) scaling of a method with the number
of electrons
• Lack of size extensivity implies errors from the exact
energy increase as more electrons enter the
calculation
Electrons
21
Size consistent
• “Non-interacting limit” description
• “Additive separability” of the wavefunction
• Entire PES must be correctly described as we bring
two non-interacting H2 molecules close together
H2
2 H2
Ea
=
r=∞
Eb
H2
22
Variational principle
• Choose an arbitrary wavefunction and vary it until
you find the lowest energy
• Boundedness by ground state energy, E0
*

 H dr
 
*
 E0
23
Accuracy
• Gold standard in QM: CCSD(T)/aug-cc-pwcv(5+d)z-DK
…method/basis set (more later)
 Ea
RT
Expt.
k  Ae
kcalc
0.1 
 10
kexpt
Theory
• Ideally, want calculated and experimental rate
constants to be within one order of magnitude from
each other
• Therefore, at 298K, theoretical energy should be
within one kcal/mol of experimental value
24
Solving the Schrodinger equation
• Can be solved exactly for a one-electron system
• Start adding in more electrons…
• Initial assumption: Pretend electrons don’t interact (Vee=0)
– Then Hamiltonian is separable and total electronic wavefunction
Ψ(r1,r2) describing motion of two electrons is product of the two
wavefunctions, Ψ(r1) Ψ(r2)
• Hartree Product
HP (r1 , r2 ,..., rN )  1 (r1 )2 (r2 )...N (rN )
• Fails antisymmetry principle
• A wavefunction describing fermions should be antisymmetric
with respect to the interchange of any set of space-spin
coordinates
25
Solving the Schrodinger equation
For two electrons…
1
1 1 ( x1 )
1 ( x1 )  2 ( x2 )  1 ( x2 )  2 ( x1 ) 
 ( x1 , x2 ) 
2
2 1 ( x2 )
 2 ( x1 )
 2 ( x2 )
If you put two electrons in same orbital at same time…
( x1 , x2 )  0
For N electrons…
1

N!
1 ( x1 )
1 ( x 2 )
Pauli Exclusion Principle!
 2 ( x1 ) ...  N ( x1 )
 2 ( x2 ) ...  N ( x2 )
...
...
...
...
1 ( xN )  2 ( xN ) ...  N ( xN )
Slater determinant
26
Solving the Schrodinger equation
Simplifying the Hamiltonian
Hˆ  
electrons

i
1 2 electrons nuclei e 2 Z A electrons e 2 nuclei e 2 Z A Z B
i   
 

2
riA
rij A B rAB
i
A
i j
Hˆ elec   h(i)   v(i, j )  VNN
i
i j
• Next up Hartree-Fock and self-consistent field!
27
Approximate solutions to Schrödinger equation
Schrödinger’s equation is nearly impossible to solve, so approximate
methods are used.
2
æ
Ñ
ç -å i - å Z I + å 1
ç
è i 2 i,I R I - r̂i i¹ j r̂i - r̂j

HF

1 1
 2 1

ö
÷ Y ( ri ) = EY ( ri )
÷
ø
1 2  1 N 
 2 2   2 N 

 N 1  N 2   N N 
Slater determinant: Use wavefunction of
noninteracting electrons for interacting system
0 r   0 r1 , r2 , r3 ,, rN   E
Density functional theory: The ground-state
electron density contains all information in the
ground-state wavefunction
• Electron repulsion makes this
problem difficult
• Approach 1: Use approximate
wavefunction forms (QM)
• Approach 2: Use the electron density
as the main variable (DFT)
• Approach 3: Approximate the energy
using empirical functions (MM)
E ( R I ) ® kIJ ( RIJ - RIJ0 )
2
Molecular mechanics: Use empirical
functions and parameters to describe the
energy (e.g. harmonic oscillator for chemical
28
bond)