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1、Basic Hartree-Fock TheoryIn Hartree-Fock (HF) theory the energy of a system is given as a sum of five components:EHF = ENN + ET + Ev + Ecoul + Eexch The nuclear-nuclear repulsion ENN describes the electrostatic repulsion between the nuclei and is independent of the electron coordinates. In time-inde
2、pendent HF theory, the kinetic energy of the nuclei is not, but the kinetic energy of the electrons ET is considered. Together with the nuclear-electron attraction energy Ev it depends on the coordinates of one electron. The classical electron-electron Coulomb repulsion energy Ecoul and the non-clas
3、sical electron-electron exchange energy Eexch 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
4、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 atomi
5、c 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 electro
6、nic 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 op
7、timization 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 wavef
8、unction 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 perform
9、ed in HF energy calculations will be illustrated using formaldehyde in its C2v 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=tightHF/STO-3G formaldehyde0
10、1C1O2 1 r2H3 1 r3 2 a3H4 1 r3 2 a3 3 180.0r2=1.21672286r3=1.10137241a3=122.73666566After 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 expa
11、nded. 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 function
12、s 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 1s2s2px2py2pz orbitals of carbon and oxygen. Overall this amounts to 12 basis functions. As each of thes
13、e 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 a
14、re 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 fun
15、ctions 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: 0.1Aside 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 ENN in atomic units (Hartree). Subsequen
