computer programs are written to solve the schrödinger equation 2 max born (german, 1882-1970)...
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Hartree-Fock Theory
Patrick TamukongNorth Dakota State UniversityDepartment of Chemistry & BiochemistryFargo, ND 58108-6050 U.S.A.June 10, 2015
Computer programs are written to solve the Schrödinger equation
EΨΨH ˆ
2
|||| AiiAiA Rrrr
|||| jiijij rrrr
|||| BAABAB RRRR
nneeenne V
M
1A
M
AB AB
BA
V
N
1i
N
ij ij
V
N
1i
M
1A iA
A
T
M
1A
2A
A
T
N
1i
2i R
ZZr1
rZ
2M1
21
H
Max Born (German, 1882-
1970)
Julius Robert Oppenheimer (Berkeley- Los alamos, 1904 –1967)
Szabo, A. Ostlund, N. S. McGraw-Hill, New York, 1989, p. 40.
Electronic Structure Problem
Born-Oppenheimer Approximation
The approximation used in solving the electronic structure problem
nneeenne V
M
1A
M
AB AB
BA
V
N
1i
N
ij ij
V
N
1i
M
1A iA
A
T
M
1A
2A
A
T
N
1i
2i R
ZZr1
rZ
2M1
21
H
=0 constant
N
i
N
i
N
ij ij
M
A iA
AN
iielec rr
ZH
1 111
2 1
2
1
M
A
M
AB AB
BAelectot R
ZZEE
1elecelecelecelec EH
Aielecelec Rr ;
})({2
1
})({2
1
1
2
1
2
1
1
2
11
2
11 1 1 1 1
22
Apottot
M
AA
A
M
A
M
AB AB
BAAelec
M
AA
A
M
A
M
AB AB
BAM
A
N
i
N
i
M
A
N
i
N
ij ijiA
AiA
Anucl
REM
R
ZZRE
M
R
ZZ
rr
Z
MH
Szabo, A. Ostlund, N. S. McGraw-Hill, New York, 1989, p. 40. 3
Atomic UnitsChosen for convenience such that (me=1, e=1, = h/2 = 1, ao=1, and the potential energy in the hydrogen atom (e2/ao = 1).
re
4π1
Δ2m
2
0
2
r
1
2
1
Other frequently used energy units:
1a.u. = 27.212 eV = 627.51 Kcal/mol = 2.1947·105 cm-1
1Kcal/mol = 4.184KJ/molBoltzmann’s constant: k = 1.38066·10-23J/K
Avogadro’s number: NA= 6.02205·1023mol-1
Rydberg constant: R∞= 1.097373·107m-1
Compton wavelength of electron: λC= 2.426309·10-
12m
Stefan-Boltzmann constant: σ = 5.67032·108W/(m2K4)
ΦEΦrε4π
e2m 0
22
e
2
ΦEΦrε4π
e2m 0
22
e
2
2
a0
2
2e
2
Eε4π
em
λλ
02e
20 aem
ε4π
4
Our Main Concern Static Electron
Correlation
Dynamic Electron Correlation
5
1*u
1gz
2g
1uyz
1uxz 5sσσ4d5sσπ4dπ4d 2
The need to include more than one electron configuration in the description of the total wave function
2YY2
contributes some 80% to the total wave function at 2.80 Å but only 55% at 4.4 Å
The need to account for the coulomb hole
Electron Correlation in He
6Helgaker et al. Molecular Electronic-Structure Theory, John Wiley & Sons Ltd, Chichester, England, 2000, p. 257.
HF Approximation
Full Treatment
Full Treatment - HF
Correlation Effect
7Tamukong, P. K.; Theis, D.; Khait, Y. G.; Hoffmann, M. R. J. Phys. Chem. A 2012, 116, 4590.
MCSCF GVVPT2
Cr2 Ground State
Conceptual Picture
RHF
RMP2Higher
MCSCF
8
Hartree-Fock Theory lacks electron correlation and represents the total wave function as a single electron configuration
Use of Molecular Orbitals A MO is a wavefunction associated with a single
electron. The use of the term "orbital" was first used by Mulliken in 1925.
MO theory was developed, in the years after valence bond theory (1927) had been established, primarily through the efforts of Friedrich Hund, Robert Mulliken, John C. Slater, and John Lennard-Jones. The word orbital was introduced by Mulliken in 1932. According to Hückel, the first quantitative use of MO theory was the 1929 paper of Lennard-Jones. The first accurate calculation of a molecular orbital wavefunction was that made by Charles Coulson in 1938 on the hydrogen molecule. By 1950, MO were completely defined as eigenfunctions of the self-consistent field
Robert Sanderson Mulliken 1996-1986Nobel 1966
Friedrich Hund 1896-1997
Charles Alfred Coulson 1910-1974
Sir John Lennard-Jones 1894-1954
9
John Clarke Slater
1900-1976
LCAO = MO
It was introduced in 1929 by Lennard-Jones with the description of bonding in the diatomic molecules of the first main row of the periodic table, but had been used earlier by Pauling for H2
+.
10
Sir John Lennard-Jones 1894-1954
Linus Carl Pauling 1901-1994 Nobel 1962
iν
K
1ννi Cψ
ωβrψ
ωαrψxχ
i
ii
The Variational Principle Method of choice for approximate solutions to
physical problems The raison d'être for the LCAO Define an approximate solution (with same
boundary conditions as the eigenvectors of the Hamiltonian) to the Shrodinger equation as a LC of its unknown eigenvectors
ααα ΦεΦH α
αα CΦΦ~
0εΦHΦ ~ˆ~then
The task then is to determine the optimal set of expansion coefficients which is accomplished by Lagrange’s method of undetermined multipliers
11
Lagrangian Method In the Lagrangian method, a function (or functional) is minimized (or maximized) subject to given equality constraints Nrdrρ
α
β α
*βαβ CCΦΦ1ΦΦ ~~
Example: One wishes to minimize subject to
yxyx,f 23yx 22
Construct the Lagrangian
3yxλyxλy,x,Λ 222
Solve the equations
03yx2yλx2xλ2xy
0λ
λy,x,Λ
y
λy,x,Λ
x
λy,x,Λ
222
2x
3y
and y = ±1
ORand x = 0
12
The Secular Equation Use the above philosophy
αββαβαβαβ
αβα
N21
1CCΦΦEΦHΦCC
1ΦΦEΦHΦE,C,,C,CΛ
ˆ
~~~ˆ~
α
αα CΦΦ~
E,C,,C,CΛ N21 Consider small variations in
0C
Λ
C
Λ
21
β β
βαββαβ CSECH where βααβ ΦHΦH ˆ
βααβ ΦΦS HC = ESC
All quantum chemistry methods solve this secular equation using different approximations. It is a matrix problem and reduces to that of matrix diagonalization. Often it is transformed into an eigenvalue problem 13
Matrix Eigenvalue Problem
Transform the secular equation into an eigenvalue problem by rewriting C as
14
CSC
2
1
HenceECSSSCHSS
2
1†
2
1
2
1†
2
1
HC = ESC
ECCH Eigenvalue problem
ISSS
2
1†
2
1
where
The above transformation is known as the symmetric (Löwdin) orthonormalization
The thus obtained matrix eigenvalue problem is the final problem solved in quantum chemistry
Hartree Approximation The total Hamiltonian is approximated as a sum of
one-electron operators and the wave function as a product of eigenvectors of those operatorsc
N
1i
N
ij1j
ij
N
1ii J
2
1HE
ii
M
1A iA
A2iiiiiii (i)dτΦ
r
Z
2
1(i)Φ(i)dτΦ(i)hΦH
212j
12
2iij dτ(2)dτΦ
r
1(1)ΦJ
)(r)...Φ(r)Φ(rΦ)r,...,r,(rΨ Nn2j1iN21HP
iiHH
The variational principle then leads to
15
E=εi+εj+…+εn
The Problem)(r)...Φ(r)Φ(rΦr,...,r,rΨ Nn2j1iN21
HP )(
Φi – spin orbitals• The form of ΨHP suggests the independence of Φi • Probability density given by ΨHP is equal to the product of monoelectronic probability densities • This is true only if each electron is completely independent of the other electrons• ΨHP - independent electron model
A ♥ A♥
PA=1/13 P♥=1/4 PA♥=1/52=PAP♥
PA is uncorrelated (independent) with P♥.
Uncorrelated probabilitiesCorrelated probabilitiesIn a n-electron system of electrons the motions of the electrons is correlated due to the Coulomb
repulsion (electron-one will avoid regions of space occupied by electron two).
E=εi+εj+…+εn
Electronic Hamiltonian can be rewritten:
ee
N
1iiE VhH
i2ii v
21
h
Where:
is the monoelectronic operator
N
1i
v
M
1A iA
AN
1iieN
i
rZ
vV
vi is the monoelectronic term of the external potential:
In HP, hi will act only on the wavefunction corresponding to the i-th electron. However, Vee depends on pairs of electrons so that we can not separate the variables in Schrödinger equation.
16
Slater Determinants The Hartree product ignores electron correlation
completely and ignores Pauli’s exclusion principle To fix this, the wave function is often written as a
slater determinant or their linear combination
NNN2N1
2N2221
1N1211
N21
xχxχxχ
xχxχxχ
xχxχxχ
N!
1x,,x,xΨ
ωβrψ
ωαrψxχ
i
ii
0
0)()()()(
1
1)()()()(
**
**
dd
and
dd
N is a normalization factor
17
In Hartree-Fock Theory, the wave function is a single slater determinant
Hartree-Fock Theory Write the Hamiltonian as a sum of Fock
operators
18
i
ifH HFi
M
1A iA
A2ii v
r
Z
2
1f
ECCF iν
K
1νν
iν
K
1ννi CφεCf
where the Hartree-Fock potential is defined in terms of coulomb and exchange operators
b
bbHFi iKiJv
2χ2χr1χ1χdxdx1χ1J1χ bb1
12aa21aba Coulomb integral
2χ2χr1χ1χdxdx1χ1K1χ ab1
12ba21aba Exchange integral
ωβrψ
ωαrψxχ
i
ii
Use the variational and langrangian
methods to arrive at
Douglas Rayner Hartree English
1897-1958
Vladimir Aleksandrovich Fock Russian 1898–1974