Linker IT Software
Google
Web www.oraxcel.com
menubar-top-links menubar-top-rechts
Home Help Search Login
Welcome, Guest. Please Login.
SQL*XL: Database to Excel bridge litLIB: Excel power functions pack ExcelLock: Locking and securing your valuable Excel spreadsheets encOffice: Protect your Excel file easy and safe encOffice: Protect your Excel file easy and safe
Pages: 1
Hartree-Fock theory (Read 9550 times)
Gerrit-Jan Linker
YaBB Administrator
*****




Posts: 75
Hartree-Fock theory
24.12.07 at 13:13:50
 
Hartree-Fock theory
 
Hartree-Fock is a mean field theory in which each electron has its own wavefunction (orbital), which in turn obeys an effective 1-electron Schrodinger equation.
 
The effective hamiltonian (Fock operator) contains the average field (Coulomb and exchange) of all other electrons in the system. In other words, the electron electron repulsion is treated in a mean field way.  
 
The total electronic wavefunction for the molecule, ignoring complications introduced by the Pauli principle, is a simple product of the orbitals.  
 
Following the Born interpretation of wavefunctions, this implies that if the probability density for finding electron 1 at position r1 is P(r1) and if the probability density for finding electron 2 at position r2 is P(r2), that the probability density for finding electron 1 at r1 and electron 2 at r2 is:
P(r1,r2) = P(r1)P(r2)
 
In other words, the probability density for a given electron is independent of the positions of all other electrons. In reality however the motions of electrons are more intimately correclated. Because of the direct Coulomb repulsion of electrons the instantaneous position of electron 2 forms a region in space that electron 1 will avoid. This avoidance is more than that caused by the mean field. So P(r1,r2) near r1=r2 is too high since electron 1 has no knowledge of the instantaneous position of electron 2. It only eperiences a field due to the average value of the position of electron 2.
 
This effect is also called electron correlation. Hartree-Fock theory neglects electron correlation effects. The correlation energy is the difference between the Hartree-Fock limit and the exact energy. A certain amount of electron correlation is already considered within the HF approximation, found in the electron exchange term describing the correlation between electrons with parallel spin. This basic correlation prevents two parallel-spin electrons from being found at the same point in space and is often called Fermi correlation.
 
HF wavefunctions  are  not  adequate  for  describing  the  formation  and  dissociation  of covalent  bonds.  The problem is that the two  electrons associated with a particular bond generally associate with different fragments (radicals) as the bond is broken. In  HF  wavefunctions  a  bond  pair  consists  of one  doubly  occupied  orbital  and hence will not, in general, lead to the correct dissociated radical species.  
 
See also:
HF band gap
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1300859601/0#0
 
Sources:
AB INITIO METHODS FOR ELECTRON CORRELATION IN MOLECULES, Peter Knowles
http://www.fz-juelich.de/nic-series/Volume3/knowles.pdf
 
See also:
http://www.eng.fsu.edu/~dommelen/quantum/style_a/hf.html
Back to top
 
« Last Edit: 01.06.12 at 07:48:14 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
Email WWW Gerrit-Jan Linker   IP Logged
Gerrit-Jan Linker
YaBB Administrator
*****




Posts: 75
Re: Hartree-Fock theory
Reply #1 - 24.07.08 at 22:35:30
 
Quote:
While Slater (1951) was examining how to speed up HF calculations he was aware that one consequence of the Pauli principle is that the Fermi exchange hole is larger than the correlation hole, i.e. , exchange corrections tot he classical interelectronic repulsuion are significantly larger than correlation corrections (typically between one and two orders of magnitude)

 
Source: Essentials of Computational Chemistry, 2nd editions Christopher J. Cramer.
 
See also:
Coulomb hole
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1212592000
Fermi hole
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1212592128
Back to top
 
« Last Edit: 24.07.08 at 22:39:36 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
Email WWW Gerrit-Jan Linker   IP Logged
Gerrit-Jan Linker
YaBB Administrator
*****




Posts: 75
HF and the independent electron approximation
Reply #2 - 27.08.09 at 07:56:49
 
HF and the independent electron approximation
 
In HF theory one electron is considered to move in the average field of all the other electrons. The HF Hamiltonian is the Schrodinger equation for 1 electron:
 
HHF = Σih(i) + Σi<j rij-1
 
Note that DFT is also a method in which the independent electron approximation is used.
 
See also:
Independent particle model
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1198616987
Back to top
 
« Last Edit: 27.08.09 at 12:41:27 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
Email WWW Gerrit-Jan Linker   IP Logged
Gerrit-Jan Linker
YaBB Administrator
*****




Posts: 75
Localised description is obtained with HF
Reply #3 - 13.07.11 at 12:30:17
 
Localised description is obtained with HF
 
In HF a state is always described with a single determinant. There are situations where a state can only be described by a CSF that is a combination of more determinants.  
 
A localised description is obtained with Hartree Fock for the reason that partial occupations are not possible in HF.  
In situations of degeneracy HF can only describe the system with a single determinant.
In some situations a single determinant is not a CSF. If e.g. a CSF = sum of 2 det's then this is not possible in HF. Only with a CI.
Back to top
 
 

Gerrit-Jan Linker
Linker IT Software
Email WWW Gerrit-Jan Linker   IP Logged
Gerrit-Jan Linker
YaBB Administrator
*****




Posts: 75
Hartree-Fock theory: Fock operator
Reply #4 - 30.10.13 at 10:53:23
 
Hartree-Fock theory: Fock operator
 
The Fock operator in spin-restricted HF theory is defined as:  
F=h+∑i(2Ji-Ki)
 
F : the Fock operator
h : one electron operator
Ji : Coulomb operator  
Ki : Exchange operator
 
The sum is over half the electrons (all alpha electrons e.g.).
 
Examples:
He 1s2 : F = h + 2J1s - K1s
Be 1s22s2 : F = h + 2J1s - K1s + 2J2s - K2s
Back to top
 
« Last Edit: 30.10.13 at 10:57:15 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
Email WWW Gerrit-Jan Linker   IP Logged
Gerrit-Jan Linker
YaBB Administrator
*****




Posts: 75
HF theory at different restraint levels
Reply #5 - 04.09.14 at 15:57:12
 
HF theory at different restraint levels
 
RHF: spin-restricted Hartree-Fock
RRHF: RHF in real coordinate space
CRHF: RHF in a complex coordinate space
UHF: spin-unrestricted Hartree-Fock
CUHF: UHF in a complex coordinate space
GHF: Generalised Hartree-Fock where no pure alpha and beta spin functions are used.
RGHF: GHF in real coordinate space
CGHF: GHF in a complex coordinate space
 
Source:
D. Dehareng, D. Dive, Journal of Computational Chemistry, Vol 21, No 6, 483 (2000)
Back to top
 
« Last Edit: 04.09.14 at 16:01:22 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
Email WWW Gerrit-Jan Linker   IP Logged
Pages: 1