Lecture 14 Beyond Hartree-Fock
A better way to do it
©2013, Jordan, Schmidt & Kable
Lecture 14
Copyright Notice
Some images used in these lectures are taken, with permission, from
“Physical Chemistry”, T. Engel and P. Reid, (Pearson, Sydney, 2006);
denoted “ER” throughout the lectures
and other sources as indicated, in accordance with
the Australian copyright regulations.
©2013, Jordan, Schmidt & Kable
Lecture 14
Learning outcomes
14.1 Can compare common basis sets and arrange in order of
increasing quality
14.2 Discuss Variational theorem in context of single
determinant and multideterminant wavefunctions.
14.3 Explain why Hartree-Fock wavefunction cannot describe
bond cleavage
14.4 Explain how to improve the Hartree-Fock wavefunction
using the N-electron basis of configurations
14.5 Recognize terms such as CI, SCF, HF
14.6 Interpret output of quantum chemistry software
©2013, Jordan, Schmidt & Kable
Lecture 14
Hamiltonian operator is like a matrix
Vectors are changed (rotated and stretched/shrunk) by matrices. If we
represent the wavefunction of a vector is some basis function space, then
the Hamiltonian operator acts like a matrix.
k
y

=
Ĥ 
=
+
Ey
k
j
i
+

= E

j
i
Where the vector is a solution to the Schrödinger equation, it will be only
stretched or shrunk upon operation with the Hamiltonian. In this case it is
called an eigenvector.
©2013, Jordan, Schmidt & Kable
Lecture 14
The Slater determinant
Generally, for a closed-shell molecule:
y1 (1)
(
)
Y 1,2....N =
1
N
y1 ( 2)
y1 (1)
y N /2 (1)
y1 ( 2)
y N /2 ( 2)
y1 ( N ) y1 ( N )
y N /2 ( N )
Exchanging two electrons is like exchanging rows, which changes the
sign of the determinant. If the third electron went into y1, then two
rows would be the same, and the determinant is zero.
©2013, Jordan, Schmidt & Kable
Lecture 14
The Hartree-Fock Hamiltonian
The Hamiltonian is the sum of kinetic and potential energies:
2
Z
e
Ĥ = å Ñi2 + å å
-å å J
i=1 2me
i=1 j=i 4pe0rij i=1 J =1 4pe0 riJ
N
2
N N
e2
N M
j j = å a jk c k
k
The basis coefficients ajk are optimized to minimize the energy of the oneelectron orbitals. But, since the Hamiltonian depends on the wavefunctions
of the other electrons, this must be iterated until “self-consistency” is
achieved.
Hartree-Fock (HF) is a Self-Consistent Field (SCF) Method.
©2013, Jordan, Schmidt & Kable
Lecture 14
The Hartree-Fock Approximation
The electronic wavefunction is written as a single Slater determinant.
y1 (1)
(
)
Y 1,2....N =
y1 ( 2)
1
N
y1 (1)
y N /2 (1)
y1 ( 2)
y N /2 ( 2)
y1 ( N ) y1 ( N )
y N /2 ( N )
The spin-orbitals are products of spatial orbitals and spin functions:
y j (i ) = j j (i ) a (i )
y j (i ) = j j (i ) b (i )
The spatial orbitals are constructed from basis functions, c.
j j = å a jk c k
k
©2013, Jordan, Schmidt & Kable
Lecture 14
The Hartree-Fock Hamiltonian
The Hamiltonian is the sum of kinetic and potential energies:
2
Z
e
Ĥ = å Ñi2 + å å
-å å J
i=1 2me
i=1 j=i 4pe0rij i=1 J =1 4pe0 riJ
N
2
N N
e2
N M
j j = å a jk c k
k
The basis coefficients ajk are optimized to minimize the energy of the oneelectron orbitals. But, since the Hamiltonian depends on the wavefunctions
of the other electrons, this must be iterated until “self-consistency” is
achieved.
Hartree-Fock (HF) is a Self-Consistent Field (SCF) Method.
©2013, Jordan, Schmidt & Kable
Lecture 14
The Hartree-Fock Method
HF procedure obtains optimized orbitals according to the variational
principle. The total wavefunction is a single Slater determinant.
{ck }
Ĥ
©2013, Jordan, Schmidt & Kable
j j = å a jk c k
k
{e j }
E
Lecture 14
Example – N2 using minimal STO-3G basis
The STO-3G basis set is the smallest that one would ever use. It has just
the core and valence orbitals represented, and is pretty much only good for
qualitative calculations.
Core and valence orbitals are
ATOMIC BASIS SET
---------------sums of three gaussians.
THE CONTRACTED PRIMITIVE FUNCTIONS HAVE BEEN UNNORMALIZED
THE CONTRACTED BASIS FUNCTIONS ARE NOW NORMALIZED TO UNITY
SHELL TYPE PRIM
EXPONENT
2p orbitals are made by
multiplication with x, y or z to
create node.
CONTRACTION COEFFICIENTS
N
3
3
3
S
S
S
1
2
3
99.106169
18.052312
4.885660
4
4
4
L
L
L
4
5
6
3.780456
0.878497
0.285714
3.454881 (
3.341410 (
1.041372 (
0.154329)
0.535328)
0.444635)
-0.193164 ( -0.099967)
0.258372 ( 0.399513)
0.194997 ( 0.700115)
TOTAL NUMBER OF SHELLS
TOTAL NUMBER OF BASIS FUNCTIONS
NUMBER OF ELECTRONS
CHARGE OF MOLECULE
STATE MULTIPLICITY
NUMBER OF OCCUPIED ORBITALS (ALPHA)
NUMBER OF OCCUPIED ORBITALS (BETA )
TOTAL NUMBER OF ATOMS
©2013, Jordan, Schmidt & Kable
=
=
=
=
=
=
=
=
1.171553 (
0.736704 (
0.116706 (
0.155916)
0.607684)
0.391957)
4
10
14
0
1
7
7
2
Lecture 14
Example – N2 using minimal STO-3G basis
The coefficients of these 3-Gaussian basis functions are optimized.
ITER EX DEM TOTAL ENERGY
E CHANGE
DENSITY CHANGE
ORB. GRAD
1 0 0 -107.370987008 -107.370987008
0.438830221
0.000000000
---------------START SECOND ORDER SCF--------------2 1 0 -107.494559234
-0.123572227
0.118946520
0.063938252
3 2 0 -107.500374882
-0.005815648
0.033282163
0.013749873
4 3 0 -107.500654278
-0.000279396
0.000295497
0.000288385
5 4 0 -107.500654311
-0.000000032
0.000001456
0.000014352
6 5 0 -107.500654311
0.000000000
0.000000145
0.000000320
INTEGRALS
1050
SKIPPED
117
1050
1050
1050
1041
1015
117
117
117
122
128
-----------EIGENVECTORS
------------
1
2
3
4
5
6
7
8
9
10
N
N
N
N
N
N
N
N
N
N
1
1
1
1
1
2
2
2
2
2
S
S
X
Y
Z
S
S
X
Y
Z
1
-15.5063
AG
0.703182
0.012857
0.000000
0.000000
0.001709
0.703182
0.012857
0.000000
0.000000
-0.001709
2
-15.5050
B1U
0.702819
0.025712
0.000000
0.000000
0.009237
-0.702819
-0.025712
0.000000
0.000000
0.009237
©2013, Jordan, Schmidt & Kable
3
-1.4085
AG
-0.173700
0.500000
0.000000
0.000000
0.230275
-0.173700
0.500000
0.000000
0.000000
-0.230275
4
-0.7275
B1U
-0.172555
0.746615
0.000000
0.000000
-0.252769
0.172555
-0.746615
0.000000
0.000000
-0.252769
5
-0.5486
B3U
0.000000
0.000000
0.629644
0.000000
0.000000
0.000000
0.000000
0.629644
0.000000
0.000000
1
2
3
4
5
6
7
8
9
10
N
N
N
N
N
N
N
N
N
N
1
1
1
1
1
2
2
2
2
2
S
S
X
Y
Z
S
S
X
Y
Z
6
-0.5486
B2U
0.000000
0.000000
0.000000
0.629644
0.000000
0.000000
0.000000
0.000000
0.629644
0.000000
7
-0.5303
AG
0.069560
-0.399585
0.000000
0.000000
0.604238
0.069560
-0.399585
0.000000
0.000000
-0.604238
8
0.2653
B2G
0.000000
0.000000
0.822656
0.000000
0.000000
0.000000
0.000000
-0.822656
0.000000
0.000000
9
0.2653
B3G
0.000000
0.000000
0.000000
0.822656
0.000000
0.000000
0.000000
0.000000
-0.822656
0.000000
10
1.0408
B1U
-0.124828
1.094636
0.000000
0.000000
1.162969
0.124828
-1.094636
0.000000
0.000000
1.162969
Lecture 14
Example – N2 using minimal STO-3G basis
…and here are your orbitals.
©2013, Jordan, Schmidt & Kable
Lecture 14
Example – N2 using minimal STO-3G basis
Here is the MO diagram
Energy (Hartree)
2
1
0
-1
-2
-15
©2013, Jordan, Schmidt & Kable
Lecture 14
Gaussian basis sets
+
+
=
STO-3G
A single basis function constructed from 3 gaussians
represents each core and valence orbital.
6-31G
Core orbitals of heavy atoms represented with 6 gaussians.
Valence orbitals represented by 3 gaussians. One extra set
of extra (smaller) valence functions made with one
gaussian.
6-311G
As above, but an extra set of more diffuse gaussians.
6-311+G
Another set of very diffuse functions for heavy atoms.
6-311++G
Add some diffuse functions for the hydrogens as well.
Polarization functions may also be added to atoms – e.g. d orbitals on
carbon and p orbitals on hydrogen. 6-31G(d) is popular, for instance.
©2013, Jordan, Schmidt & Kable
Lecture 14
Variational Theorem: Energies of H atom
Common basis sets show that the larger the basis, the lower the energy of
the H atom. The correct energy is approached with the largest basis set.
Energy (hartree)
-0.46
-0.48
Correct energy is -0.5 Eh
-0.50
STO-3G
6-31G
6-311G
Basis Set
©2013, Jordan, Schmidt & Kable
6-311++G
Error is only 40cm-1 at 6-311++G
Lecture 14
Hartree-Fock Potential energy curve of H2
We use the 6-311++G(d,p) basis set.
-0.90
-0.92
-0.94
-0.96
2 H atoms = -1.0 Eh
Energy (hartree)
-0.98
-1.00
-1.02
-1.04
-1.06
-1.08
-1.10
-1.12
-1.14
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
H-H distance (Å)
Somehow, Hartree-Fock does not describe dissociation properly…. Why?
©2013, Jordan, Schmidt & Kable
Lecture 14
Hartree-Fock wavefunction for H2
HF wavefunction is antisymmetrized product of 2 1s sigma spin-orbitals.
( )
Y HF 1,2 =
y1s (1) y1s ( 2) - y1s (1) y1s (1))
(
2
1
()
()
= j1s 1 j1s 2
Symmetric space function
j1s =
©2013, Jordan, Schmidt & Kable
() ( ) () ( )
éa 1 b 2 - b 1 a 2 ù
ë
û
2
Antisymmetric spin function
j1sa + j1sb )
(
2
1
Lecture 14
Hartree-Fock wavefunction for H2
HF spatial wavefunction is simple product of 2 1s sigma orbitals.
(
2
j1s (1) j1s ( 2) =
=
1
(
j1sa (1) + j1sb (1)
()
()
) 2 (j1sa (2) + j1sb (2))
1
()
()
()
()
( ))
()
1
j1sa 1 j1sa 2 + j1sa 1 j1sb 2 + j1sb 1 j1sa 2 + j1sb 1 j1sb 2
2
As the molecule dissociates, these terms describe both electrons being on
the one atom, which is H- plus a proton! This adds a coulombic attraction
term which prevents proper dissociation.
What we have not accounted for is electron CORRELATION
©2013, Jordan, Schmidt & Kable
Lecture 14
Hartree-Fock wavefunction for H2
Look at the wavefunction as a 2d plot along bond axis
Ionic terms
1.0
(1,2)
0.8
0.6
0.4
0.2
4
2
0.0
-2
0
z
1
©2013, Jordan, Schmidt & Kable
2
z
-2
2
0
-4
-4
4
Lecture 14
The set of configurations are an N-electron basis
If a single determinant wavefunction is not enough, then add
more determinants! The various configurations form an
orthonormal basis in the N-electron space.
k
y

=
Ĥ 
=
+
Ey
k
j
i
+

= E

j
i
Here i, j, and k are determinants which represent different configurations.
Calculations which allow the interaction of configurations is called…..
Configuration Interaction (CI).
©2013, Jordan, Schmidt & Kable
Lecture 14
Configuration Interaction for H2
Adding in (with a negative coefficent) some of the doubly excited
configuration can remove ionic terms.
Ionic terms
1.0
1.0
0.8
0.6
0.8
0.2
0.6
(1,2)
(1,2)
0.4
0.4
0.0
-0.2
-0.4
-0.6
0.2
-0.8
1
2
©2013, Jordan, Schmidt & Kable
z
-4
1
4
j1s (1) j1s ( 2)
-2
0
2
-2
0
z
-2
z
-2
0
-4
0
-4
1s*
1s
2
z
0.0
4
2
-1.0
2
4
2
-4
4
j1s * (1) j1s * ( 2)
1s*
1s
Lecture 14
Configuration Interaction for H2
We do a complete active space self-consistent field (CAS-SCF) theory
calculation, to allow both configurations while simultaneously optimizing the
orbitals, with the same basis as before.
At 0.9Å
ITER
1
2
3
4
5
TOTAL ENERGY
-1.125049420
-1.141745458
-1.141768656
-1.141768724
-1.141768726
DEL(E)
LAG.ASYMM. SQCDF MICIT
-1.125049420 0.160266 1.377E-02 1
-0.016696038 0.004000 1.652E-04 1
-0.000023198 0.000164 7.577E-07 1
-0.000000069 0.000020 1.318E-08 1
-0.000000001 0.000003 2.313E-10 1
DAMP
0.0000
0.0000
0.0000
0.0000
0.0000
-------------------LAGRANGIAN CONVERGED
-------------------FINAL MCSCF ENERGY IS
-1.1417687256 AFTER
5 ITERATIONS
j1s (1) j1s ( 2)
1s*
1s
j1s * (1) j1s * ( 2)
-MCCI- BASED ON OPTIMIZED ORBITALS
----------------------------------
1s*
1s
CI EIGENVECTORS WILL BE LABELED IN GROUP=D2H
PRINTING ALL NON-ZERO CI COEFFICIENTS
STATE
1
ENERGY=
-1.1417687256
S=
0.00
SZ=
0.00
SPACE SYM=AG
ALPH|BETA| COEFFICIENT
----|----|-----------10 | 10 |
0.9900962
01 | 01 | -0.1403909
HF energy was -1.11832. CASSCF energy is -1.14177 Eh
©2013, Jordan, Schmidt & Kable
Variational
Principle!!
Lecture 14
Configuration Interaction for H2
-0.90
-0.92
-0.94
-0.96
Energy (hartree)
-0.98
-1.00
-1.02
-1.04
ALPH|BETA| COEFFICIENT
----|----|-----------10 | 10 |
0.8803840
01 | 01 | -0.4742616
-1.06
-1.08
-1.10
-1.12
ALPH|BETA| COEFFICIENT
----|----|-----------10 | 10 |
0.9947259
01 | 01 | -0.1025694
-1.14
-1.16
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
H-H distance (Å)
CI wavefunction decribes dissociation (chemistry!) because it accounts for CORRELATION.
Wavefunction at equilibrium is well described by a single configuration (determinant).
©2013, Jordan, Schmidt & Kable
Lecture 14
Level of correlation (# of configurations)
The Quantum Chemistry Landscape
The answer
pointless
Sensible
compromise
Hartree-Fock limit
1st year chemistry
Quality of basis set
©2013, Jordan, Schmidt & Kable
Lecture 14