Atomic units
The atomic units have been chosen such that the fundamental electron properties are all equal to one
atomic unit. (me=1, e=1, = h/2 = 1, ao=1, and the potential energy in the hydrogen atom (e2/ao = 1).
1D=3.33564·10-30Cm
The use of atomic units also simplifies Schrödinger's equation. For example the Hamiltonian for an electron in the
Hydrogen atom would be:
2
1 e2


2m
4 0 r
1
1
 
2
r
Other fundamental constants:
Boltzmann’s constant:
k=1.38066·10-23J/K
Avogadro’s number:
NA=6.02205·1023mol-1
Rydberg constant:
1a.u. = 27.212eV = 627.51Kcal/mol = 2.1947·105 cm-1
R∞=1.097373·107m-1
Compton wavelength of electron:
λC=2.426309·10-12m
1Kcal/mol = 4.184KJ/mol
Stefan-Boltzmann constant:σ=5.67032·108W/(m2K4)
Other frequently used energy units:
Approximations made in Hartree-Fock-Roothaan-Hall theory
Basic concepts, techniques and notations of molecular quantum mechanics
 structure of many-electron operators (e.g. Hamiltonian)
 form of many-electron wave-functions (Slater determinants, and linear combination of them)
 Hartree-Fock (HF) approximation
 more sophisticated approaches which use the HF method as a starting point
The electronic problem
The non-relativistic time-independent Schrödinger equation
H|Φ>=E|Φ>
H – Hamiltonian operator for a system of nuclei and electrons

 
riA | riA || ri  RA |

 
rij | rij || ri  r j |
R AB



| R AB || R A  RB |
N M
Z A N N 1 M M Z AZB
1 2 M 1
2
H    i  
 A  
    
2
2M
r
i 1
A 1
i 1 A 1 i A
i 1 ji ri j
A 1 B A R AB


 
A 

 

 


 
N
Te
A molecular coordinate system
Tn
Ven
Vee
(1)
Vnn
MA - the ratio of the mass of nucleus A to the mass of an electron
ZA – the atomic number of nucleus A
Te – the operator for the kinetic energy of the electrons
Tn – the operator for the kinetic energy of the nuclei
Vee– the operator for the Coulomb attraction between electrons and nuclei
Vee – the operator for the repulsion between electrons
Vnn – the operator for the repulsion between nuclei
(1) – represents the general problem
 to be separated in two parts: electronic and nuclear problems
Born-Oppenheimer Approximation
The nuclei are much heavier than electrons
 they move much more slowly
 the nuclei can be considered frozen in a single arrangement (molecular conformation)
 the electrons can respond almost instantaneously to any change in the nuclear position
► the electrons in a molecule are moving in the field of fixed nuclei
► 2-nd term in (1) can be neglected
► 5-th term in (1) is a constant
N M
Z A N N 1 M M Z AZB
1 2 M 1
2
H    i  
 A  
    
constant
2
2M
r
i 1
A 1
i 1 A 1 i A
i 1 ji ri j
A 1 B A R AB


 
A 

 

 


 
N
=0
Electronic Hamiltonian
Te
Tn
describes the motion of N electrons in the field of M point charges
Vee
Vnn
N
H elec
1 2 N M ZA N N 1
   i   
 
2
r
i 1
i 1 A1 iA
i 1 j i rij
Ven
Electronic Schrödinger equation:
Helec|Φelec>=Eelec|Φelec>
(3)
Φelec=Φelec({rI};{RA}) (4)
Eelec = Eelec({RA})
(5)
(2)
(4) - is the electronic wave-function which describes the motion of the
electrons
explicitly depends on the electronic coordinates parametrically
depends on the nuclear coordinates
parametric dependence
the nuclear coordinates do not appear explicitly in Φelec.
different wave-function is defined for each nuclear configuration
The total energy:
M
M
Z AZ B
A1 B  A R AB
Etot  Eelec   
(6)
Equations (2) – (6) ≡ electronic problem
If the electronic problem is solved
► we can solve for the motion of the nuclei
Since the electrons move much faster than the nuclei
► we can replace the electronic coordinates by their average values (averaged over the electronic wave-function)
 nuclear Hamiltonian
o
describes the motion of the nuclei in the average field of the electrons
M
H nucl
N
M M
Z Z
1
1 2 N M ZA N N 1
2
 
 A     i  
 
 A B
A1 2 M A
i 1 2
i 1 A1 riA
i 1 j i rij
A1 B  A R AB
M
M M
Z Z
1
2
 
 A E elec ({R A })    A B
A1 2 M A
A1 B  A R AB
M
1
 2A Etotpot ({R A })
A1 2 M A
 
potential energy surface (PES)
pot
Etot
({RA})
nuclear Schrödinger equation
Hnucl|Φnnucl> = E|Φnucl>
Φnucl - describes the vibration, rotation and translation of a molecule
E - total energy of the molecule(in the Born-Oppenheimer approximation)
- includes: - electronic energy
- vibrational energy
- rotational energy
- translational energy
Schematic illustration of a potential energy surface
The equilibrium conformation of the molecule
corresponds to the minimum of the surface
Total wave-function in Born-Oppenheimer approximation:
Φ({ri};{RA}) = Φelec({ri};{RA})·Φnucl({RA})
Born-Oppenheimer approximation
- usually a good approximation
- bad approximation for:
 excited states
 degenerate or cuasidegenerate states
The Antisymmetry or Pauli Exclusion Principle
electron spin
α(ω) and β(ω) –spin functions (complete and orthonormal)

*
( ) ( )d    * ( )  ( )d  1
     1
and

*
( )  ( )d    * ( ) ( )d  0
     0
the electron is described by spatial (r) and spin (ω) coordinates:
x={r,ω}
A many electron wave-function must be antisymmetric with respect to the interchange of the
coordinate x (both space and spin) of any two electrons.
Φ(x1, x2, ... , xi, ..., xj, ...,xN) = -Φ(x1, x2, ... , xj, ..., xi, ...,xN)
Hartree Approximation (Hartree, 1928)
Ψ HP (r1 , r2 ,..., rN )  Φ i (r1 )Φ j (r2 )...Φ n (rN )
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 probabilities
Correlated probabilities
In 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:
N
H E   h i  V ee
i 1
Where:
1
h i    2i  v i
2
is the monoelectronic operator
vi is the monoelectronic term of the external potential:
Z
VeN   v i    A
ri A
i 1
i 1 A 1



N
N
M
vi
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.
Hartree Approximation: the electrons do not interact explicitly with the others, but each electron
interacts with the medium potential given by the other electrons
Using the variational methods one obtains the energy of the system:
N
1 N N
E   H i   J i j
2 i 1 j 1
i 1
j i
where:
 1 2 M ZA 
H i   Φ i (i)h i Φ i (i)dτ i  Φ i (i)   i   
Φ i (i)dτ i
2
r
A 1
iA 

1
J i j   Φ 2i (1) Φ 2j (2)dτ 1 dτ 2
r12
- core monoelectronic integrals
-Coulombian bielectronic integrals
-represent the classical repulsion energy between two charge densities described by Φi and Φj
Bielectronic potential 1/r12 felt by the electron 1, due to the instantaneous position of electron 2 is replaced by a
monoelectronic potential Vi(1) obtained by averaging the interaction between the two electrons over the spatial and spin
coordinates of electron 2. Summing over j≠i one obtains the medium potential acting on electron in Φ i and which is due to
the other N-1 electrons
N
Vi (1)    Φ j (2)
j 1
j i
Coulomb operator
J j (1)   Φ j (2)
2
2
1
dτ 2
r12
1
dτ 2 represents the local medium potential felt by electron 1 and due to the electron described by Φj
r12
2 1


J j (1) i (1)   Φ j (2)
dτ 2  i (1)
r12


Using the Lagrange’s multipliers method  Hartree equations:


M
N N
2 1
Z
1
  2   A 
Φ j (j)
dτ j Φ i (i)  ε i Φ i (i)


i

 2

ri A i 1 j1
ri j
A 1


j i
N
ε i  Hi   Jij
- the energies of molecular orbitals
j 1
j i
Total electronic energy:
N
N -1
N
i 1
i 1 j  i  1
E   i    Jij
In order to find Φi we need Φi  SCF procedure
SCF procedure in the framework of Hartree approximation
ρ i (r)  Φ i (r)
2
electronic density corresponding to the i-th electron
N
N
i 1
i 1
ρ tot (r)   ρ i (r)   Φ i (r)
2
total electronic density
Each electron interacts with an electronic density obtained by subtracting its density from the total density
ρ
(k)
N
N

2
2
(r)  ρ tot (r)  ρ k (r)    ρ i (r)  Φ k (r)   Φ i (r)
i 1
 i 1

ik
Vee potential can be written as:
N
Vee   g i (r)
i 1
with:
1
g i (r)   ρ (i ) (r' ) dr'
r
gi(r) - interaction energy of the point charge (the considered individual electron) with the other electrons represented
as an electronic density
Hamiltonian:
 1

H el      2i  v i  g i 

i 1  2
N
Hartree equations:
 1 2




v

g
i
i
i
 2
 Φ i (i)  ε i Φ i (i)


Determinantal wave-functions: Hartree-Fock approximation
ΨHP - does not satisfy the Pauli principle
- gives a non-zero probability for two electrons to be exactly at the same point in space
Fock, Slater, 1930
Φ i (r1 ) Φ j (r1 ) ... Φ n (r1 )
Φ i (r2 ) Φ j (r2 ) ... Φ n (r2 )
Ψ SD (r1 , r2 ,..., rN )  (N! ) 1/2




Φ i (rN ) Φ j (rN ) ... Φ n (rN )
ΨSD antisimetrized sum of Hartree products with all the possible distributions of the electrons in the molecular orbitals
Ψ SD (r1 , r2 ,..., rN )  Φ i (r1 )Φ j (r2 )...Φ N (rN )
- shorthand notation
Using the variational method of Ritz:
N
E  Ψ SD H Ψ SD   H i 
i 1
N
N
1 N N
E   H i   J i j
2 i 1 j 1
i 1
N
1
(J i j  K i j)

2 i1 j1
j i
 1

H i   Φ *i (r)h i Φ i (r)dτ   Φ *i (r)  2i  v i Φ i (r)dτ
 2

K i j   Φ*i (r1 )Φ*j (r1 )
1
Φ i (r2 )Φ j (r2 )dr1 dr2
r1  r2
exchange integral
In Hartree
approximation
exchange operator:


1
K j (1)Φi (1)   Φ j (2) Φi (2)dτ 2 Φ j (1)
r12


- a non-local operator because its result depends on the value of Φi on entire space and not only on the value of Φi
where is located the electron 1
JijKij0
Minimizing the energy by varying the spin orbitals leads to the Hartree-Fock equations:
N
N
2
1
1
 1 2

*



v
Φ
(r)

Φ
(r'
)
Φ
(r)dτ

Φ
(r'
)Φ
(r'
)
Φ j (r)dτ j   i  i (r)


i
i
i
j
i
j
j
i


 2

r
r'
r
r'
j 1
j 1
Definig the Fock operator:
N
f i  h i   (J j  K j )
molecular orbital energies:
j 1
N
ε i  Φ i f i Φ i  H i   (J i j  K i j )
j 1
total electronic energies:
total energy:
N
1 N N
E   i  (J i j  K i j)
2 i1 j1
i 1
N
N -1
N
E   i    Jij
i 1
i 1 j  i  1
In Hartree
approximation
M M
Z Z
1 N N
E T    i   (J i j  K i j )    A B
2 i 1 j 1
i 1
A 1 B  A R AB
N
if we use the spatial orbitals:
Φ1(x)=φ1(r)α(ω) şi Φ2(x)=φ1(r)β(ω)
 1 (r1 ) (s 1 )  1 (r1 ) (s 1 )
 1 (r2 ) (s 2 )  1 (r2 ) (s 2 )
SD
1/2
Ψ RHF (r1 , r2 ,..., rN )  (N! )


 1 (rN ) (s N )  1 (rN ) (s N )
...  n (r1 ) (s 1 )
...  n (r2 ) (s 2 )


...  n (rN ) (s N )
in the framework of RHF approximation:
N/2
N/2 N/2
i 1
i 1 j1
E  2 Hi   (2J i j  K i j )
molecular orbital energies:
N/2
 i  H i   (2J i j  K i j )
j 1
N/2
N/2 N/2
i 1
i 1 j 1
E  2  i   (2J i j  K i j )
Hartree-Fock equations = alternative Schrödinger equation
in which the exact Hamiltonian has been replaced by an
approximate Fock operator
- Coulomb operator has been replaced by an operator
which describes the interaction of each electron with the
average field due to the other electrons
RHF and UHF formalisms
{φi}, i=1,...k
Given a set of k orthonormal spatial orbitals (MO)
 2k spin-orbitals: Φi,
i=1,...,2k
 2i 1 ( x)   i (r ) ( )

 2i ( x)   i (r )  ( ) 
i  1, K
restricted OM  restricted wave-function
RHF  1s1s 2 s
Restricted wave-function for Li atom
But: K1s()2s( )≠0 and K1s()2s()=0
1s() and 1s() electrons will experience different potentials so that it will be more
convenient to describe the two kind of electrons by different wave-functions
UHF  1s 1s  2s
Unrestricted wave-function for Li atom
 iα  jα  δ ij
 iβ  βj  δ i j
 iα  βj  S αβ
ij
usually, the two sets of spatial orbitals use the same basis set
UHF wave-functions are not eigenfunctions of S2 operator !!!
 spin contamination
2
  c 22 2  c 42 4  c 62 6  ...
|2> - exact doublet state
-approximately a singlet
- approximately a doublet
|4> - exact quartet state
|6> - exact sextet state
For an UHF wave-function, the expectation value of S2 is:
S
2
UHF
 S
exact
N
N
 N   S

2
i
2

ij
j
where:
S
2
exact
S2
 N  N 
 
2

UHF
 N   N 


 1
2


 S2
exact
spin projection procedures (Gaussian)
 iα  jα  δ ij
 iβ  βj  δ i j
 iα  βj  S αβ
ij