Chapter 3
The
Computational
Problem
The Computational Problem
3.1
3.1 Typical Program Structure
Specify geometry
Determine symmetry
Specify basis set
Calculate integrals
Calculate an initial guess
Central
Program
Mass
Storage
Solve Roothaan’s equation
or
Kohn-Sham equations
Population Analysis
Calculate gradients
Calculate frequencies
The Computational Problem
3.2
Sequence of Execution
Specify geometry
Determine Symmetry
Specify basis set
Calculate integrals
Generate an initial guess
Solve equations
Population analysis
.
.
.
The Computational Problem
3.3
Typical Sequence of Execution for Geometry Optimization
Specify geometry
Determine Symmetry
Specify basis set
Calculate integrals
Generate an initial guess
Calculate integrals
Solve equations
Calculate gradient
Converged?
Yes
No
Predict new geometry
The Computational Problem
3.4
MUNgauss structure (Dynamic execution)
Initialization
Object required
Object1
Object2
Object4
Output ! This is how an object could be printed
Object=CLASS:NAME%MODALITY
End
Object3
Object5
The Computational Problem
3.5
3.2 Molecular Geometry
All programs require Cartesian coordinates (integrals, gradients, ...)
Most common way of defining a molecular structure is the
Z-matrix (internal coordinates).
Spartan uses only Cartesians, but building a molecule is easy.
Can also use WebMO or GaussView
The Z-matrix is converted to Cartesian
Cartesian can be converted to Proper or Natural Internal
coordinates(PIC, NIC) or Redundant Internal Coordinates(RIC)
(for optimization purposes only).
The Z-matrix (MUNgauss ≈ Gaussian):
First atom defines the origin (0, 0, 0)
Second atom defines the z-axis (0, 0, Z)
Third atom defines the x-z plane (X, 0, Z)
Fourth ... atoms have (X, Y, Z)
NOTE:
May be minor differences between programs
Gaussian uses RIC by default
Examples ...
The Computational Problem
Torsion angles (dihedral angle)
3.6
The Computational Problem
3.7
H2O (C2v symmetry)
atom atom distance atom angle atom torsion +/N1
N2
N3
Euler
O1
D*2 O1
1.0
H3
O1
OH
D*2 HOD
H4
O1
OH
D*2 HOD H3
180.0
blank line
OH = 0.96
HOD = 127.5
blank line ----------------------------------------------------------------D* (MUNgauss only) or X (Gaussian or MUNgauss) represent dummy
atoms
N.B. Have imposed C2v symmetry
The Computational Problem
3.8
A. NH2F (Cs symmetry)
atom atom distance atom angle atom torsion +/N1
N2
N3
Euler
N1
F2
N1
NF
D*3 N1
1.0
F2
H4
N1
NH
D*3 HND F2
+90.0
H5
N1
NH
D*3 HND F2
-90.0
DNF
blank line
NF = 1.387
NH = 1.049
DNF = 113.5
HND = 51.6
blank line -----------------------------------------------------------------
The Computational Problem
3.9
B. NH2F (Cs symmetry)
atom atom distance atom angle atom torsion +/N1
N2
N3
Euler
N1
D*2 N1
1.0
F3
N1
NF
D*2 FND
H4
N1
NH
D*2 HND F3
+DH1
H5
N1
NH
D*2 HND F3
-DH1
blank line
NF = 1.387
NH = 1.049
FND = 114.0
HND = 110.0
DH1 = 122.0
blank line -----------------------------------------------------------------
The Computational Problem
3.10
CH3OH (Cs) using torsion angles
atom atom distance atom angle
N1
N2
atom torsion +/N3
Euler
C1
O2
C1
RCO
H3
O2
ROH
C1
COH
H4
C1
RCH4
O2
OCH4 H3
180.0
H5
C1
RCH
O2
OCH
H4
+TOW
H6
C1
RCH
O2
OCH
H4
-TOW
blank line
RCO = 1.43
ROH = 0.96
RCH4 = 1.09
RCH = 1.09
COH = 105.0
OCH4 = TETRA ! (MUNgauss only)
OCH = TETRA
TOW = 120.0
blank line -----------------------------------------------------------------
The Computational Problem
3.11
CH3OH (Cs) using Euler angles
atom atom distance atom angle
N1
N2
atom torsion +/N3
Euler
C1
O2
C1
RCO
H3
O2
ROH
C1
COH
H4
C1
RCH4
O2
OCH4 H3
180.0
H5
C1
RCH
O2
OCH
H4
TOW
+1
H6
C1
RCH
O2
OCH
H4
TOW
-1
blank line
RCO = 1.43
ROH = 0.96
RCH4 = 1.09
RCH = 1.09
COH = 105.0
OCH4 = TETRA ! (MUNgauss only)
OCH = TETRA
TOW = TETRA
blank line -----------------------------------------------------------------
The Computational Problem
3.12
3.3 Geometry Optimization
Born-Oppenheimer → Potential Energy Curve
or Potential Energy Surface
E = E(R) = function of nuclear coordinates
Want to find Re (Equilibrium geometry) (structure)
At Re, E = minimum and therefore,
For H2O, could have R1, R2, and ∝, therefore,
and want
In general,
= Potential Energy Surface (PES)
Two basic choices:
The Computational Problem
- cartesian coordinates n = 3N (or 3N-6)
- internal coordinates n = 3N-6 (usually)
Bonds; angles, torsions example, z-matrix, RIC, PIC
PES
-can have many stationary points
-stationary points can be
minima = equilibrium structures
saddle points (first order) = transition state structures
NOTE: RIC >>3N-6 coordinates
3.13
The Computational Problem
Type of stationary points
1.
Energy minimum = equilibrium structure (reactants, products,
intermediates)
2.
Saddle points (≥ 1 imaginary frequencies)
(a) first order = transition state structure
i.e., one imaginary frequency
(b) Higher order (≥ 2)
≥2 imaginary frequencies “no chemical interest”
3.14
The Computational Problem
Example of a simple Potential Energy Surface (PES)
3.15
The Computational Problem
3.16
Optimization Methods (Algorithms)
Assume a quadratic surface,
Using bra-ket notation,
(1)
Differentiating equation (1)
Solving for |q〉,
(2)
Let |q〉* = stationary point, |g〉 = 0,
(3)
(3)-(2)
The Computational Problem
or,
Finding Equilibrium Structures
Iterative process = Geometry optimization
∣q〉i+1
∣q〉i
∣g〉i
= coordinates at iteration i+1
= coordinates at iteration i
= gradients at iteration i
3.17
The Computational Problem
H = Hessian = approximation to G (Force constant matrix)
Usually use an approximate Hessian which is updated during the
optimization.
Optimization Methods
Minimization:
1. Energy (point by point)
2. Gradient based
3. Hessian based
4. Simulated annealing
5. Genetic Algorithms
Choice of H
1) H = G, Newton’s or Newton-Raphson
*2) H ≈ G, Quasi-Newton (variable metric)
*a) Steepest-descent, H=1
b) DFP (Davidon-Fletcher-Powell)
*c) BFGS (Broyden-Fletcher-Goldfarb-Shanno)
*d) Murtagh-Sargent-Powell (MSP)
*e) OC (Optimally Conditioned method by Davidon)
*3) DIIS (Direct Inversion of the Iterative Subspace)
Can update H, H-1, J (H=JJ+)
Transition states require different optimization methods (not a
minimization)
* Available in MUNgauss
Gaussian uses the Bernie method
3.18
The Computational Problem
Approximate Hessian used as initial guess:
Cbasis and DAB are empirical constants, RAB is the actual bond length.
3.19
The Computational Problem
3.20
Interpretation:
(a) Small Force Constant
is large
large step
(b) Large Force Constant
small step
is small
The Computational Problem
3.21
Types of Stationary Points (revisited)
G = Force constant matrix
1. Minimum
G has all positive eigenvalues (positive definite)
2. Maximum
G has all negative eigenvalues
3. Saddle points
G has n negative eigenvalues
nth order saddle point, n ≥ 1
Note: Difference between “Transition state” and “Transition structure”
The Computational Problem
Transition State Structure Optimization
BFGS and OC are formulated to ensure a minimum.
MUNgauss options:
1. Constrained optimization (rarely possible)
2. Minimization of Sum of Squares (VA05AD)
For a stationary point, f → 0
Uses numerical differentiation:
Choice of coordinate more important for transition states
Gaussian:
OPT=(TS, CalcFC, NoEigenTest)
3.22
The Computational Problem
Reaction Paths
Is the steepest descent path from a transition structure down to
reactants and down to products.
F. Fukui, Acc. Chem. Res., 14, 363 (1981).
Depends on choice of coordinate system:
The Intrinsic Reaction Coordinate (IRC), uses mass-weighted
cartesian coordinates,
3.23
The Computational Problem
3.24
3.4 Direct Methods
The direct SCF
In direct methods (e.g., direct SCF), the integrals are re-evaluated
at every iteration.
For larger calculations (larger number of basis functions, K), it may
be impossible to store all integrals on disk.
Normally, can have a combination of,
some integrals kept in core (memory),
some integrals kept on disk and,
some re-evaluated (direct).
The Computational Problem
3.25
3.5 Symmetry Properties
Consider H2O
C2v point group
Reflected in the Hamiltonian and therefore the wavefunction
M.O. (and vibrational modes ...) belong to irreducible representations
(I.R.) of the corresponding point group.
C2v point group
I.R.
a1, a2, b1 and b2
Electronic configuration (ground state) for H2O
Important for post-HF calculations
Spectroscopy
Speeding up calculations
Excited States