basic HF theory

Hartree-Fock Theory

In Hartree-Fock (HF) theory the energy of a system is given as a sum of five components:

E_HF = E_NN + E_T + E_v + E_coul + E_exch

The nuclear-nuclear repulsion E_NN describes the electrostatic repulsion between the nuclei and is independent of the electron coordinates. In time-independent HF theory, the kinetic energy of the nuclei is not, but the kinetic energy of the electrons E_T is considered. Together with the nuclear-electron attraction energy E_v it depends on the coordinates of one electron. The classical electron-electron Coulomb repulsion energy E_coul and the non-classical electron-electron exchange energy E_exch depend on the coordinates of two electrons. Calculation of these last two terms constitutes the main effort in HF calculations and is also responsible for the unfavorable formal scaling of computational effort as the fourth power of basis functions used for the description of the wavefunction. The particular assumption made in HF theory is that each electron feels the other electrons only as an average charge cloud, but not as individual electrons.

The molecular electronic wavefunction in HF theory is based on the LCAO (=linear combination of atomic orbitals) scheme describing each molecular orbital (holding one electron) as a linear combination of basis functions (usually located at the nuclear center):

The molecular orbital coefficients describe the contribution of each of the basis functions to a given molecular orbital. The overall electronic wavefunction of the system is constructed as an antisymmetrized product of the molecular orbitals (the Slater determinand) in order to fulfill the Pauli exclusion principle. Since the optimal shape of one molecular orbital depends on the shape of all the other occupied molecular orbitals, the optimization of the overall wavefunction is achieved in an iterative manner, varying the molecular orbital coefficients until no further changes in the overall wavefunction occur. The direction of the variation of MO coefficients is guided by the variational principle stating that an approximate wavefunction yields an energy of the system which is higher than the energy obtained from the exact wavefunction. In other words: in order to arrive at the most favorable wavefunction the MO coefficients must be varied such that the energy of the system becomes as low as possible.

The basic steps performed in HF energy calculations will be illustrated using formaldehyde in its C_2v structure as example. Please observe that the output obtained from HF calculations will vary dramatically depending on whether normal (#) or extendet (#P) output is specified:

#P HF/STO-3G scf=tight

HF/STO-3G formaldehyde

0 1
C1
O2  1  r2
H3  1  r3  2  a3
H4  1  r3  2  a3  3  180.0

r2=1.21672286
r3=1.10137241
a3=122.73666566

After reading in the geometry of the system and determining its symmetry properties, the actual first step of Hartree-Fock calculations consists in choosing a basis set in which the electronic wavefunction can be expanded. This is done in link 301 of the program, usually by calling a predefined set of basis functions from a library. In the current case we are using the standard STO-3G basis set containing three Gaussian functions (primitives) to describe each Slater-type "atomic" orbital. Sufficient basis functions are used to describe the core and valence orbitals of the system. For formaldehyde there will be one basis function for the 1s orbitals of hydrogen and five basis functions each to describe the 1s2s2p_x2p_y2p_z orbitals of carbon and oxygen. Overall this amounts to 12 basis functions. As each of these basis functions is described by three Gaussian type functions, we have 36 primitive Gaussians for the system.

 (Enter /scr1/g03/l301.exe)
 Standard basis: STO-3G (5D, 7F)
 There are     7 symmetry adapted basis functions of A1  symmetry.
 There are     0 symmetry adapted basis functions of A2  symmetry.
 There are     2 symmetry adapted basis functions of B1  symmetry.
 There are     3 symmetry adapted basis functions of B2  symmetry.
 Integral buffers will be    262144 words long.
 Raffenetti 1 integral format.
 Two-electron integral symmetry is turned on.
    12 basis functions,    36 primitive gaussians,    12 cartesian basis functions
     8 alpha electrons        8 beta electrons
       nuclear repulsion energy        31.0872308470 Hartrees.
 IExCor=   0 DFT=F Ex=HF Corr=None ExCW=0 ScaHFX=  1.000000
 ScaDFX=  1.000000  1.000000  1.000000  1.000000
 IRadAn=      0 IRanWt=     -1 IRanGd=            0 ICorTp=0
 NAtoms=    4 NActive=    4 NUniq=    3 SFac= 2.05D+00 NAtFMM=   60 Big=F
 Leave Link  301 at Mon Nov 22 15:27:09 2004, MaxMem=    6000000 cpu:       0.1

Aside from basis set considerations link 301 also reports the number of electrons (here 16 electrons; 8 alpha and 8 beta spin electrons) and the nuclear repulsion energy E_NN in atomic units (Hartree). Subsequent links 30x (x=2,3, . . ) are responsible for the computation of one and two electron integrals.

 (Enter /scr1/g03/l302.exe)
 NPDir=0 NMtPBC=     1 NCelOv=     1 NCel=       1 NClECP=     1 NCelD=      1
         NCelK=      1 NCelE2=     1 NClLst=     1 CellRange=     0.0.
 One-electron integrals computed using PRISM.
 One-electron integral symmetry used in STVInt
 NBasis=    12 RedAO= T  NBF=     7     0     2     3
 NBsUse=    12 1.00D-06 NBFU=     7     0     2     3
 Leave Link  302 at Mon Nov 22 15:27:10 2004, MaxMem=    6000000 cpu:       0.5
 (Enter /scr1/g03/l303.exe)
 DipDrv:  MaxL=1.
 Leave Link  303 at Mon Nov 22 15:27:11 2004, MaxMem=    6000000 cpu:       0.0

A first guess for the wavefunction of the system is made in link 401. While earlier versions of Gaussian used the semiempirical INDO method to derive a first guess of the wavefunction, the Harris density functional method is used more recently. The orbital symmetries are printed for the occupied and the empty (virtual) orbitals.

 (Enter /scr1/g03/l401.exe)
 Harris functional with IExCor=  205 diagonalized for initial guess.
 ExpMin= 1.69D-01 ExpMax= 1.31D+02 ExpMxC= 1.31D+02 IAcc=1 IRadAn=         1 AccDes= 1.00D-06
 HarFok:  IExCor= 205 AccDes= 1.00D-06 IRadAn=         1 IDoV=1
 ScaDFX=  1.000000  1.000000  1.000000  1.000000
 Harris En= -112.417169397061
 Initial guess orbital symmetries:
       Occupied  (A1) (A1) (A1) (A1) (B2) (A1) (B1) (B2)
       Virtual   (B1) (A1) (B2) (A1)
 The electronic state of the initial guess is 1-A1.
 Leave Link  401 at Mon Nov 22 15:27:12 2004, MaxMem=    6000000 cpu:       0.3

The actual Hartree-Fock energy calculation is performed by link 502 in an iterative manner until self consistency of the wavefunction (self consistent field (SCF) calculation) is achieved. At the beginning of link 502 the convergence criteria (rms and max. density changes) are listed together with the default number of iterations (here 64). The most important information listed for each SCF cycle are the actual energy of the electronic subsystem (that is, the total energy without the core-core repulsion energy) as e.g. E= -112.352697484853, and the energy change with respect to the previous SCF iteration, e.g. Delta-E= -0.017123861810. All energies are given in atomic units.

 (Enter /scr1/g03/l502.exe)
 Closed shell SCF:
 Requested convergence on RMS density matrix=1.00D-08 within 128 cycles.
 Requested convergence on MAX density matrix=1.00D-06.
 Requested convergence on             energy=1.00D-06.
 No special actions if energy rises.
 Using DIIS extrapolation, IDIIS=  1040.
 Integral symmetry usage will be decided dynamically.
 Keep R1 integrals in memory in canonical form, NReq=      422883.
 IEnd=     18774 IEndB=     18774 NGot=   6000000 MDV=   5995234
 LenX=   5995234
 Symmetry not used in FoFDir.
 MinBra= 0 MaxBra= 1 Meth= 1.
 IRaf=       0 NMat=   1 IRICut=       1 DoRegI=T DoRafI=F ISym2E= 0 JSym2E=0.

 Cycle   1  Pass 1  IDiag  1:
 E= -112.335573623044
 DIIS: error= 3.88D-02 at cycle   1 NSaved=   1.
 NSaved= 1 IEnMin= 1 EnMin= -112.335573623044     IErMin= 1 ErrMin= 3.88D-02
 ErrMax= 3.88D-02 EMaxC= 1.00D-01 BMatC= 1.97D-02 BMatP= 1.97D-02
 IDIUse=3 WtCom= 6.12D-01 WtEn= 3.88D-01
 Coeff-Com:  0.100D+01
 Coeff-En:   0.100D+01
 Coeff:      0.100D+01
 Gap=     0.612 Goal=   None    Shift=    0.000
 GapD=    0.612 DampG=2.000 DampE=0.500 DampFc=1.0000 IDamp=-1.
 RMSDP=1.40D-02 MaxDP=6.46D-02              OVMax= 0.00D+00

 Cycle   2  Pass 1  IDiag  1:
 E= -112.352697484853     Delta-E=       -0.017123861810 Rises=F Damp=F
 DIIS: error= 1.00D-02 at cycle   2 NSaved=   2.
 NSaved= 2 IEnMin= 2 EnMin= -112.352697484853     IErMin= 2 ErrMin= 1.00D-02
 ErrMax= 1.00D-02 EMaxC= 1.00D-01 BMatC= 1.32D-03 BMatP= 1.97D-02
 IDIUse=3 WtCom= 9.00D-01 WtEn= 1.00D-01
 Coeff-Com:  0.502D-01 0.950D+00
 Coeff-En:   0.000D+00 0.100D+01
 Coeff:      0.452D-01 0.955D+00
 Gap=     0.643 Goal=   None    Shift=    0.000
 RMSDP=4.61D-03 MaxDP=1.69D-02 DE=-1.71D-02 OVMax= 0.00D+00

.
.
.
.

 Cycle   9  Pass 1  IDiag  1:
 E= -112.354347141204     Delta-E=       -0.000000000001 Rises=F Damp=F
 DIIS: error= 1.42D-08 at cycle   9 NSaved=   9.
 NSaved= 9 IEnMin= 9 EnMin= -112.354347141204     IErMin= 9 ErrMin= 1.42D-08
 ErrMax= 1.42D-08 EMaxC= 1.00D-01 BMatC= 2.72D-15 BMatP= 4.82D-13
 IDIUse=1 WtCom= 1.00D+00 WtEn= 0.00D+00
 Coeff-Com: -0.374D-07 0.186D-05 0.127D-04-0.740D-04 0.477D-03-0.300D-02
 Coeff-Com:  0.194D-01-0.164D+00 0.115D+01
 Coeff:     -0.374D-07 0.186D-05 0.127D-04-0.740D-04 0.477D-03-0.300D-02
 Coeff:      0.194D-01-0.164D+00 0.115D+01
 Gap=     0.636 Goal=   None    Shift=    0.000
 RMSDP=8.54D-09 MaxDP=3.94D-08 DE=-9.66D-13 OVMax= 0.00D+00

 SCF Done:  E(RHF) =  -112.354347141     A.U. after    9 cycles
             Convg  =    0.8544D-08             -V/T =  2.0089
             S**2   =   0.0000
 KE= 1.113601121070D+02 PE=-3.271252835904D+02 EE= 7.232359349512D+01
 Leave Link  502 at Mon Nov 22 15:27:16 2004, MaxMem=    6000000 cpu:       0.4

At convergence the total energy (now including the core-core repulsion energy) is reported as E(RHF) together with the number of SCF cycles required to reach convergence (here 9), the ratio of kinetic to potential energy -V/T , the electronic kinetic energy E_T as KE, the electron-nuclear attraction energy E_v as PE, and the sum of Coulomb and exchange interaction energies E_coul + E_exch as EE. Further information on the converged wavefunction is given by link 601:

 (Enter /scr1/g03/l601.exe)
 Copying SCF densities to generalized density rwf, ISCF=0 IROHF=0.

 **********************************************************************

            Population analysis using the SCF density.

 **********************************************************************

 Orbital symmetries:
       Occupied  (A1) (A1) (A1) (A1) (B2) (A1) (B1) (B2)
       Virtual   (B1) (A1) (B2) (A1)
 The electronic state is 1-A1.
 Alpha  occ. eigenvalues --  -20.31271 -11.12507  -1.33744  -0.80775  -0.63291
 Alpha  occ. eigenvalues --   -0.54553  -0.44319  -0.35438
 Alpha virt. eigenvalues --    0.28199   0.62863   0.73441   0.91294
          Condensed to atoms (all electrons):
              1          2          3          4
     1  C    4.736158   0.444579   0.372144   0.372144
     2  O    0.444579   7.788267  -0.022449  -0.022449
     3  H    0.372144  -0.022449   0.630937  -0.037118
     4  H    0.372144  -0.022449  -0.037118   0.630937
 Mulliken atomic charges:
              1
     1  C    0.074976
     2  O   -0.187948
     3  H    0.056486
     4  H    0.056486
 Sum of Mulliken charges=   0.00000
 Atomic charges with hydrogens summed into heavy atoms:
              1
     1  C    0.187948
     2  O   -0.187948
     3  H    0.000000
     4  H    0.000000
 Sum of Mulliken charges=   0.00000
 Electronic spatial extent (au):  =    58.6695
 Charge=     0.0000 electrons
 Dipole moment (field-independent basis, Debye):
    X=     0.0000    Y=     0.0000    Z=    -1.5369  Tot=     1.5369
 Quadrupole moment (field-independent basis, Debye-Ang):
   XX=   -10.4253   YY=   -10.5461   ZZ=   -11.3098
   XY=     0.0000   XZ=     0.0000   YZ=     0.0000
 Traceless Quadrupole moment (field-independent basis, Debye-Ang):
   XX=     0.3351   YY=     0.2143   ZZ=    -0.5494
   XY=     0.0000   XZ=     0.0000   YZ=     0.0000
 Octapole moment (field-independent basis, Debye-Ang**2):
  XXX=     0.0000  YYY=     0.0000  ZZZ=     2.6823  XYY=     0.0000
  XXY=     0.0000  XXZ=     1.6239  XZZ=     0.0000  YZZ=     0.0000
  YYZ=     0.4668  XYZ=     0.0000
 Hexadecapole moment (field-independent basis, Debye-Ang**3):
 XXXX=    -6.8834 YYYY=   -14.5710 ZZZZ=   -39.9089 XXXY=     0.0000
 XXXZ=     0.0000 YYYX=     0.0000 YYYZ=     0.0000 ZZZX=     0.0000
 ZZZY=     0.0000 XXYY=    -3.8029 XXZZ=    -7.7079 YYZZ=    -8.2855
 XXYZ=     0.0000 YYXZ=     0.0000 ZZXY=     0.0000
 No NMR shielding tensors so no spin-rotation constants.
 Leave Link  601 at Mon Nov 22 15:27:18 2004, MaxMem=    6000000 cpu:       0.8

This entails the symmetries of occupied and virtual orbitals (which may or may not be identical to the symmetries listed for the wavefunction guess), the symmetry of the electronic state of the system, orbital energies of all orbitals (in Hartree), and essential results from a Mulliken population analysis. The components of various multipoles (dipole, quadrupole . . ) are given at the end.

last changes: 22.11.2004, HZ
questions & comments to: zipse@cup.uni-muenchen.de