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Unformatted text preview: Solution of the
Electronic Schrödinger Equation
Using Basis Sets to Solve the HartreeFock Equations Comments on HartreeFock Theory
Reasonable Description of Electronic Structure
Most atoms
Many molecules near their equilibrium geometries (although there
are exceptions) Basis for More Accurate Calculations
HF is zeroorder wave function used in many more accurate models
of the electronic structure of atoms and molecules that include
electron correlation
• Configuration interaction
• Perturbation theory
• Coupled cluster theory Using Basis Sets to Solve the HartreeFock Equations HartreeFock Wave Function and Energy
Just to review, the HartreeFock (HF) wave function for Ne electrons is
e ˆ
(1,2,3,..., Ne ) = A a (1) b (2) c (3) n ( Ne ) a (2) a ( Ne ) b 1 a (1) b (2) b ( Ne ) n = (1) (1) n (2) n ( Ne ) Ne ! where i is a spin orbital, i.e., a combination of a spatial function and a spin
function. The energy of the HF wave function is: E= Ne
i =1 = Ne
i =1 h ii + Ne Ne i =1 j = i +1 h ii + 1
2 Ne Ne
i =1 j =1 (J ij K ij ) (J ij K ij ) Using Basis Sets to Solve the HartreeFock Equations HartreeFock Equations
The optimum orbitals to use in the HF wave function are the solutions of the
HF equations ˆ
f HF i ˆ
= [h + Ne
k =1 ˆ
( Jk ˆ
K k )] i = ii i where ˆ
h= 1
2 ˆ
J k = dr2
ˆ
K k = dr2 2
1 NN One electron operator:
(kinetic energy plus nuclear attraction) ZI I =1 rI 1 *
(2) k
k (2) r12
*
ˆ
(2) p12 k
k r12 (2) Coulomb operator Exchange operator ˆ
( p12 is the permutation operator) Using Basis Sets to Solve the HartreeFock Equations RHF Equations for Closed Shell Singlet States
For a singlet state with all of the orbitals doubly occupied, Ne = 2n, and the
HF wave function is
1 RHF
(1,2,3,..., Ne
e ˆ
)=A aabb nn The HF equations are ˆ
f RHF i ˆ
= h+ n
k =1 ˆ
(2 J k ˆ
Kk ) i = ii These are the closed shell Restricted HartreeFock (RHF) wave function and
equations.
For atoms and diatomic molecules, the integrodifferential RHF equations can
be solved by finite difference or finite element methods. For polyatomic
molecules, these approaches pose significant mathematical problems and the
equations have traditionally been solved using basis set expansion methods. Using Basis Sets to Solve the HartreeFock Equations Basis Set Expansion Method
The orbital i in the RHF equation is a function of a set of oneelectron
variables, (x1, y1, z1), and thus it can be represented by an expansion in a basis
set:
i where = Nbf c
=1 i is also a function of (x1, y1, z1). If the basis set, { }, is a complete set, we will obtain an exact solution of the
RHF equations. For practical reasons, it is not possible to use a complete
basis set, so an error is always associated with the use of this approach.
However, as the basis set approaches completeness, this error becomes
vanishingly small. Using Basis Sets to Solve the HartreeFock Equations Matrix HartreeFock Equations
Using the basis set expansion method, the RHF equations become ˆ
f RHF i ˆ
= f RHF Nbf
=1 Nbf ˆ
f RHF If we now project
Nbf
=1 dr1 ii = c i =1 i Nbf ci= =1
*
μ onto ci= Nbf c i i =1 these equations and integrate, we obtain *
μ
Nbf
=1 ˆ
f RHF Nbf ci= R
Fμ HF c i = i =1 dr1 *
μ c i Nbf
i =1 Sμ c i Since these equations must hold for all μ, they reduce to the matrix equation
F RHF C = S C
Using Basis Sets to Solve the HartreeFock Equations Matrix HartreeFock Equations (cont’d) In the matrix RHF equations
RHF
F11 F1RHF
K RHF
F21 RHF
F22 RHF
F2 K RHF
FK 1 F= RHF
F12 RHF
FK 2 RHF
FKK S11 S1K S21 S22 S2 K SK1 SK 2 S= S12 S KK Fock Matrix Overlap Matrix c11
C= c12 c1K 11 0 0 c21 c22 c2 K 0 22 0 cK 1 cK 2 cKK 0 0 KK Coefficient Matrix = Orbital Energy Matrix
(diagonal) Using Basis Sets to Solve the HartreeFock Equations Kinetic Energy & Nuclear Attraction Matrix Elements
The oneelectron contributions to the Fock matrix are
R
Fμ HF dr1 *
μ (r )
1 ( 1
2 ˆ
dr1 * h
μ = 2
1 ) = Tμ kinetic energy
terms + k =1 NN ( r1 ) n I =1 dr1 *
ˆ
(2 J k
μ dr1 *
μ (r )
1 ZI
rI 1 ˆ
Kk ) ( r1 ) n
= Vμ ucl nuclear attraction
terms Since there are Nbf basis functions in the basis set, Nbf 2 oneelectron integrals
of each type must be computed to construct the one electron matrix. Actually,
since the kinetic and nuclear attraction operators are hermitian, only
Nbf(Nbf+1)/2 integrals must be computed.
Using Basis Sets to Solve the HartreeFock Equations Coulomb Matrix Elements
The twoelectron Coulomb contribution to the Fock matrix is
R
Fμ HF 2 n
k =1 ˆ
dr1 * ( r1 ) J k ( r1 )
μ = ˆ
dr1 * h
μ ( r1 ) = 2
=2 = = n
k =1
n
k =1 K + dr1 K 2 K =1 =1 *
μ (r )
1
n
k =1 D dr2
K * c kc *
k ( r2 ) k ( r2 ) * K dr1dr2 c* k c dr1dr2
*
μ (r )
1 ( r2 ) dr2 k ˆ
Kk ) ( r1 ) r12 =1 =1 k *
ˆ
(2 J k
μ dr1 k =1 dr1 * ( r1 )
μ =1 =1
K n ( r2 ) ( r1 ) r12 *
μ (r )
1 * ( r1 ) ( r2 ) ( r2 ) r12
( r1 ) * r12 density matrix
Using Basis Sets to Solve the HartreeFock Equations ( r2 ) ( r2 ) = K K =1 =1 D μ Exchange Matrix Elements
The twoelectron exchange contribution to the Fock matrix is
R
Fμ HF n
k =1 ˆ
dr1 * ( r1 ) K k ( r1 )
μ = ˆ
dr1 * h
μ ( r1 ) =
= = = n
k =1
n
k =1
K + dr1 * ( r1 )
μ
dr1 n K *
μ (r )
1 K dr2 K =1 =1 1
2 D * dr1dr2 ( r1 ) r12
* K k ˆ
Kk ) *
ˆ
k ( r2 ) p12 k ( r2 ) =1 =1 c kc *
ˆ
(2 J k
μ dr1 k =1 =1 =1 k =1
K n c* k c dr1dr2
*
μ (r )
1 k dr2 ˆ
( r2 ) p12 ( r2 ) ( r1 ) r12 *
μ (r )
1 ( r1 ) * ( r2 ) ( r2 ) r12
( r1 )
r12 * ( r2 ) ( r2 ) = 1
2 K K =1 =1 D μ There are O(Nbf4/8) twoelectron integrals, [μ  ]. The cost of computing
these integrals dominates the cost of the integral calculation.
Using Basis Sets to Solve the HartreeFock Equations Matrix Elements of the Fock Operator
Combining the oneelectron and the twoelectron terms, the Fock matrix
elements become
R
Fμ HF = Tμ + Vμnucl + Nbf Nbf D
=1 =1 (μ  1
2 μ ) = Tμ + Vμnucl + Gμ
So, the problem reduces to calculating the oneelectron kinetic energy,
nuclear attraction and twoelectron 1/r12 integrals over the selected set of
basis functions, { μ}, and then solving the matrix equation
F RHF C = S C
Solution of this equation involves transformation to an orthonormal basis set
(the { μ} are usually normalized but not orthogonal) and then diagonalization
of the resulting matrix. The eigenvalues of the matrix correspond to the
orbital energies, { ii}, and the eigenvectors, transformed back to the original
basis set, correspond to the expansion coefficients, {c i}.
Using Basis Sets to Solve the HartreeFock Equations RHF Energy
The electronic contribution to the total RHF energy is
RHF
Ee =2 n
i =1 =2 n
i =1 h ii + n n i =1 j =1 (2J ij Dμ (Tμ + Vμ ) + K ij )
n n i =1 j =1 ( Dμ 2 μ  The nuclear contribution is EN = NN NN
I =1 J = I ZI ZJ
RIJ and the total energy is
RHF
RHF
Etotal = Ee + E N Using Basis Sets to Solve the HartreeFock Equations μ ) SelfConsistent Field (SCF) Procedure
There is one complication with the solution of the matrix RHF equations, the
Fock matrix elements depend on the solutions to the equations through the
density matrix, D . To address this problem, the equations are solved
iteratively:
1.
2.
3. Specify the molecule and geometry: nuclear coordinates, {RI}; nuclear charges,
{ZI}; and number of electrons (Ne).
Select the basis set, { μ}.
Calculate the integrals: Tμ , Vμ nucl, [μ  ] and Sμ . 4. Construct a guess for the density matrix, D 5.
6. Construct the Fock matrix.
Solve the matrix RHF equations, obtaining a new set of coefficients, {c 7.
8. Form a new density matrix D new.
Determine if the difference between the new and old density, D
matrix is below a set threshold. If not, return to step #5. 9. Calculate the total energy of the molecule at the given geometry. old, or expansion coefficients, {c Using Basis Sets to Solve the HartreeFock Equations new – i D i new}. old, old}. Selection of Basis Functions
We will now consider the selection of the functions to be used in basis sets for
solving the RHF equations for atoms and molecules.
There are two important criteria for selecting these functions.
1.
2. The functions must be capable of providing an accurate description of the atomic
and molecular RHF wave functions.
It must be possible to rapidly calculate the integrals Tμ , Vμ nucl, [μ  ] and Sμ
with the functions. In addition, we want the basis sets to be as compact as possible. Using Basis Sets to Solve the HartreeFock Equations Hydrogenic Orbitals and Slater Functions
The solutions of the electronic Schrödinger equation for the hydrogen atom
are:
n=2 n=1
1s = Z3 e Zr 2s 1 Z3
=
(2 Z r ) e
42 1 Zr
2 2 px 1 Z3
=
Z xe
42 1 Zr
2 etc. There are natural functions to use in the solution of the electronic
Schrödinger equation. In 1930, John C. Slater proposed a slight modification
of the radial functions in the hydrogenic orbitals, namely r n 1e r Slater functions have been used extensively in calculations on atomic and
diatomic molecules and, if the exponents ( ) are optimized, have been shown
to yield very accurate solutions of the HF equations. Unfortunately, it is very
difficult to compute the multicenter, twoelectron integrals, [μ  ], with
Slater functions. As a result, it is not feasible to use Slater functions in
calculations on polyatomic molecules.
Using Basis Sets to Solve the HartreeFock Equations Gaussian Functions
In 1950, S. F. Boys noted that all of the integrals, including the troublesome
multicenter, twoelectron integrals could be computed analytically if Gaussian
functions
l nlm mn ( x , y , z ) = N nlm x y z e r2 were used instead of Slater functions (Nnlm is a normalization constant).
The occurrence of r2 in the exponential, rather than r, causes Gaussian
functions to be less appropriate than Slater functions for use in atomic and
molecular calculations.
1. 2. Mathematically, it can be shown that the wave function should have a cusp at the
nucleus (origin). The derivative of the Gaussian function is continuous (and
zero). So, Gaussian functions will have difficulty describing the wave function
near the nucleus.
Mathematically, it can also be shown that, at large distances from the nucleus, the
wave function should fall off exponentially with r. So, Gaussian functions will
have difficulty describing the “tail” of the wave function.
Using Basis Sets to Solve the HartreeFock Equations Gaussian Functions and the Hydrogen Atom
To better understand how Gaussian functions may fare in atomic and
molecular calculations, consider their use in the hydrogen atom.
1.0x101 B
B 1.0x102 Error in Energy (h) B
3 1.0x10 Four Gaussian functions
achieve millihartree accuracy B
B 4 1.0x10 B
B 1.0x105 B Ten Gaussian functions
achieve microhartree accuracy B 1.0x106 B
B 7 1.0x10 B
B
B 8 1.0x10 1.0x109 B 0 5 n 10 Not great, but not
prohibitive! 15 Using Basis Sets to Solve the HartreeFock Equations Gaussian Functions and the Neon Atom
Let us now examine the somewhat more difficult case of the neon atom.
More functions are required to achieve comparable accuracy in this case.
1.0x100
B
sset
B
B
J
B
J
B
B
J
B
J
B J 1 1.0x10 Error in Energy (h) 2 1.0x10 1.0x103 (10s)(11s) and (6p) Gaussian
functions achieve millihartree
accuracy B
B
J
B
B
J
B
J
B
B
J
B
J
B
pset
B
J
B
J
B J 4 1.0x10 5 1.0x10 1.0x106
1.0x107 (19s) and (11p) Gaussian
functions achieve microhartree
accuracy 8 1.0x10 0 5 10 n 15 20 25 Using Basis Sets to Solve the HartreeFock Equations Gaussian Basis Sets
Much work has been done on determining the optimum Gaussian basis sets to
use in molecular HF calculations. This has resulted in a proliferation of
“recommended” basis sets. Many of these basis sets have been collected in
the Gaussian Basis Set Library at Pacific Northwest National Laboratory:
http://www.emsl.pnl.gov/forms/basisform.html In 2001, F. Jensen (Ref. 1) reported a systematic study of Gaussian basis sets
for use in molecular HF calculations. These studies defined a sequence of
basis sets that systematically approach the complete basis set limit.
To provide an accurate solution of the molecular HF equations, the basis sets
must first and foremost provide an accurate description of the atomic HF
orbitals. But, when a molecule is formed from atoms, the atomic orbitals are
perturbed. The orbitals may contract or expand. They may become polarized
toward or away from the neighboring atoms. They may even delocalize onto
nearby atoms. The basis set must be able to describe all of these effects. Using Basis Sets to Solve the HartreeFock Equations Augmentation of Gaussian Basis Sets
Contraction/Expansion of the Atomic Orbitals
If several Gaussian functions are used to describe an atomic orbital, most of
the contraction or expansion of the orbital can be accounted for by simply
varying the coefficients in the basis function expansion. Thus, expansion and
contraction can be accounted for except for the smallest atomic basis sets.
Polarization of the Atomic Orbitals
To describe polarization effects, additional functions must be added to the
basis set. We will discuss this topic next.
Delocalization of the Orbitals
Delocalization of the atomic orbitals is equivalent to building ionic character
into the wave function. If there is extensive delocalization, extra functions
may be needed to describe the ionic character of the wave function. We will
discuss this topic after we discuss polarization functions. Using Basis Sets to Solve the HartreeFock Equations Polarization Functions for Gaussian Basis Sets
When a molecule like H2 is formed, each of the atomic orbitals, in this case a
1sorbital at R = ∞, polarizes toward the other atom to help strengthen the
bond. It does this by mixing 2p character into the wave function.
+ 1s + – + 2p =
1s + 2p So, polarizing the atomic orbitals requires the addition of higher angular
momentum functions to the basis set.
Jensen (Ref. 1) also systematically investigated the polarization functions to
include in the basis sets. He did this through calculations on a set of
prototypical molecules. The (sp) and polarization sets (dfg…) were matched
by requiring that the sets give rise to identical errors (to the extent that it was
possible to do so).
Using Basis Sets to Solve the HartreeFock Equations Energy Lowerings in HF Calculations on N2
1x100 J
J 1 1x10 H 3 1x10 H Ñ Ñ 6 1x10 B F B B
B J 1x10 B
B J 0 5 h i
(18s11p6d3f2g1h) B
J 7 1x108 É
Ç B J
H Ç g B
B J
H F (13s8p4d2f1g) B
J H É f Ñ (10s6p2d1f) d B J H F 1x105 B
J H 1x104 B J p F B J
F J (7s4p1d) s H BB J 1x102 En,n1 (h) B B 10 n 15 20 Using Basis Sets to Solve the HartreeFock Equations 25 Polarization Consistent Basis Sets
Based on convergence studies of a number of prototypical molecules, Jensen
proposed the following Gaussian basis sets for use in molecular calculations: Set H C–F pc0 (3s) (5s3p) pc1 (4s1p) (7s4p1d) pc2 (6s2p1d) (10s6p2d1f) pc3 (9s4p2d1f) (14s9p4d2f1g) pc4 (11s6p3d2f1g) (18s11p6d3f2g1h) These sets are referred to as polarization consistent basis sets (pcn). Using Basis Sets to Solve the HartreeFock Equations pcn Sets: Oxygen sExponents core L L L L L L L L L LLL L L L L 2 L L L L L LLLLL L L L 1 L L 1x103 1x104 1x105 1x106 L L L L LL LL L
LLL LL Using Basis Sets to Solve the HartreeFock Equations 1x101 L 1x100 L 1x101 L 1x102 L 3 1x107 pcn 4 valence Convergence of NH Properties with pcn Sets
Re μe 1.01731 1.6133 5.51
0.55
0.007
0.000 0.0844
0.0636
0.0082
0.0006 De
HF Limita 48.73 Errorb
(7s4p1d)
(10s6p2d1f)
(14s9p4d2f1g)
(18s11p6d3f2g1h) 2.82
0.54
0.038
0.003 a J. Kobus, private communication.
b In kcal/mol, milliÅ, and Debyes. Using Basis Sets to Solve the HartreeFock Equations Convergence of CO Properties with pcn Sets
Re μe 1.10178 0.2650 4.09
0.46
0.067
0.014 0.0333
0.0244
0.0021
0.0003 De
HF Limita
183.79 Errorb
(7s4p1d)
(10s6p2d1f)
(14s9p4d2f1g)
(18s11p6d3f2g1h) 4.71
0.852
0.060
0.0006 a J. Kobus, private communication.
b In kcal/mol, milliÅ, and Debyes. Using Basis Sets to Solve the HartreeFock Equations Diffuse Functions for Gaussian Basis Sets
In 2002, Jensen (Ref. 3) explored the addition of diffuse functions to the pcn
sets developed for density functional (DF) calculations. Because of the
similarity between DF and HF basis sets, these functions can also be used to
supplement the HF basis sets also.
Addition of diffuse functions to the pcn basis sets were shown to
significantly improve the convergence of a number of molecular properties
such as the dipole and quadrupole moments and dipole polarizabilities as well
as electron affinities. The largest improvements were realized by adding a
single diffuse function to the lower angular momentum sets (s, p, and d). Using Basis Sets to Solve the HartreeFock Equations Contraction of Gaussian Basis Sets
Many of the steps in HF calculations depend on the number of functions in
the basis set. The size of the basis set can be reduced by combining the
primitive basis functions ( pprim) into contracted basis functions ( pcont):
cont
μ = L
p =1 prim
d pμ
p Although this does not reduce the amount of work required to compute the
integrals, it does reduce the work needed to diagonalize the Fock matrix.
One way to contract the basis set is to take the expansion coefficient, {dpμ},
from the atomic orbitals. This defines a minimum basis set of contracted
functions. Unfortunately, such a restricted set of contracted functions cannot
describe the impact of molecular formation on the atomic orbitals.
A very effective way to include these effects is to add the most diffuse
primitive functions in the basis set to the minimal basis set. These are the
functions that overlap the nearby atoms and provide the flexibility needed to
describe the perturbations due to molecular formation.
Using Basis Sets to Solve the HartreeFock Equations Contracted pc2 Set for Oxygen Atom: [4s3p]
1(1s) s [4s] 14989.10
2248.08
511.616
144.713
46.9056
16.6009
6.16341
1.72634
0.679615
0.236456 p [3p] 62.0205
14.3226
4.35340
1.49260
0.516428
0.170182 0.000522
0.004038
0.020725
0.080526
0.232576
0.433056
0.347467
0.041790
0.008217
0.002398
1(2p) 0.006269
0.043187
0.165515
0.361422
0.447563
0.235753 2(2s) 0.000118
0.000920
0.004746
0.019065
0.058999
0.137038
0.174532
0.167974
0.603102
0.372481 3 4 0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0 0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0 2 3 0.0
0.0
0.0
0.0
1.0
0.0 0.0
0.0
0.0
0.0
0.0
1.0 Using Basis Sets to Solve the HartreeFock Equations Contracted pc2 Set for Oxygen Atom: [2d1f]
d [2d] 1 2.3
0.65 1.0
0.0 f [1f] 0.0
1.0 1 1.2 2 1.0 The above basis set is referred to as a generally contracted basis set because a
primitive function can appear in any number of contracted functions, e.g., the
primitive s functions contribute to at least two contracted functions, 1 and 2.
If a primitive function appears in only one contracted function, the set is
referred to as a segmented contracted basis set.
Generally contracted sets are more efficient that segmented contracted sets for
molecular calculations. Nonetheless, most of the sets in the literature are
segmented sets because most integral programs cannot efficiently handle
generally contracted sets.
Using Basis Sets to Solve the HartreeFock Equations Contracted pcn Gaussian Basis Sets
Based on studies of a number of molecules, Jensen (Ref. 1) proposed the
following contracted Gaussian basis sets for use in molecular calculations: Set H C–F pc0 [2s] [3s2p] pc1 [2s1p] [3s2p1d] pc2 [3s2p1d] [4s3p2d1f] pc3 [5s4p2d1f] [6s5p4d2f1g] pc4 [7s6p3d2f1g] [8s7p6d3f2g1h] Using Basis Sets to Solve the HartreeFock Equations Convergence of NH Properties with pcn Sets
Re μe 1.01731 1.6133 De
HF Limita 48.73 Errorb
[3s2p1d]
[4s3p2d1f]
[6s5p4d2f1g]
[8s7p6d3f2g1h] 3.77
0.612
0.046
0.003 10.78
0.58
0.008
0.000 a J. Kobus, private communication.
b In kcal/mol, milliÅ, and Debyes. Using Basis Sets to Solve the HartreeFock Equations 0.0560
0.0651
0.0082
0.0006 References
1. “Polarization consistent basis sets: Principles,” F. Jensen, J. Chem. Phys. 115,
91139125 (2001). 2. “Polarization consistent basis sets. II. Estimating the KohnSham basis set limit,”
F. Jensen, J. Chem. Phys. 116, 73727379 (2002), 3. “Polarization consistent basis sets. III. The importance of diffuse functions,” F.
Jensen, J. Chem. Phys. 117, 92349240 (2002). 4. “Polarization consistent basis sets. IV. The basis set convergence of equilibrium
geometries, harmonic vibrational frequencies, and intensities,” F. Jensen, J. Chem.
Phys. 118, 24592463 (2003). 5. “Polarization consistent basis sets. V. The elements SiCl,” F. Jensen, J. Chem.
Phys. 121, 34633470 (2004). Using Basis Sets to Solve the HartreeFock Equations ...
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This note was uploaded on 12/07/2011 for the course CHEM 350 taught by Professor Duanejohnson during the Summer '06 term at University of Illinois, Urbana Champaign.
 Summer '06
 DuaneJohnson
 Equilibrium, Electron, Mole

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