Hartree-Fock principle

The problem has now become how to solve for the set of molecular orbital expansion coefficients, c. . Hartree-Fock theory takes advantage of the variational principle, which says that for the ground state of any antisymmetric normalized function of the electronic coordinates, which we will denote H, then the expectation value for the energy corresponding to E will always be greater than the energy for the exact wave function ... 

This is perhaps the easiest method to understand. It is based on the variational principle (Appendix B), analogous to the HF method. The trial wave function is written as a linear combination of determinants with the expansion coefficients determined by requiring that the energy should be a minimum (or at least stationary), a procedure known as Configuration Interaction (Cl). The MOs used for building the excited Slater determinants are taken from a Hartree-Fock calculation and held fixed. Subscripts S, D, T etc. indicate determinants which are singly, doubly, triply etc. excited relative to the... 

The best wave function of the approximate form (Eq. 11.38) may then be determined by the variational principle (Eq. II.7), either by varying the quantity p as an entity, subject to the auxiliary conditions (Eq. 11.42), or by varying the basic set fv ip2,. . ., ipN subject to the orthonormality requirement. In both ways we are lead to Hartree-Fock functions pk satisfying the eigenvalue problem... 

The correlation error can, of course, be defined with reference to the Hartree scheme but, in modem literature on electronic systems, one usually starts out from the Hartree-Fock approximation. This means that the main error is due to the neglect of the Coulomb correlation between electrons with opposite spins and, unfor-tunetely, we can expect this correlation error to be fairly large, since we force pairs of electrons with antiparallel spins together in the same orbital in space. The background for this pairing of the electrons is partly the classical formulation of the Pauli principle, partly the mathematical fact that a single determinant in such a case can... 

This theorem follows from the antisymmetry requirement (Eq. II.2) and is thus an expression for Pauli s exclusion principle. In the naive formulation of this principle, each spin orbital could be either empty or fully occupied by one electron which then would exclude any other electron from entering the same orbital. This simple model has been mathematically formulated in the Hartree-Fock scheme based on Eq. 11.38, where the form of the first-order density matrix p(x v xx) indicates that each one of the Hartree-Fock functions rplt y)2,. . ., pN is fully occupied by one electron. 

To properly describe electronic rearrangement and its dependence on both nuclear positions and velocities, it is necessary to develop a time-dependent theory of the electronic dynamics in molecular systems. A very useful approximation in this regard is the time-dependent Hartree-Fock approximation (34). Its combination with the eikonal treatment has been called the Eik/TDHF approximation, and has been implemented for ion-atom collisions.(21, 35-37) Approximations can be systematically developed from time-dependent variational principles.(38-41) These can be stated for wavefunctions and lead to differential equations for time-dependent parameters present in trial wavefunctions. 

Among the many ways to go beyond the usual Restricted Hartree-Fock model in order to introduce some electronic correlation effects into the ground state of an electronic system, the Half-Projected Hartree-Fock scheme, (HPHF) proposed by Smeyers [1,2], has the merit of preserving a conceptual simplicity together with a relatively straigthforward determination. The wave-function is written as a DODS Slater determinant projected on the spin space with S quantum number even or odd. As a result, it takes the form of two DODS Slater determinants, in which all the spin functions are interchanged. The spinorbitals have complete flexibility, and should be determined from applying the variational principle to the projected determinant. 

In the early sixties, it was shown by Roothaan [ 1 ] and Lowdin [2] that the symmetry adapted solution of the Hartree-Fock equations (i.e. belonging to an irreducible representation of the symmetry group of the Hamiltonian) corresponds to a specific extreme value of the total energy. A basic fact is to know whether this value is associated with the global minimum or a local minimum, maximum or even a saddle point of the energy. Thus, in principle, there may be some symmetry breaking solutions whose energy is lower than that of a symmetry adapted solution. 

The wave funetion obtained eorresponds to the Unrestricted Hartree-Fock scheme and beeomes equivalent to the RHF ease if the orbitals (t>a and (()p are the same. In this UHF form, the UHF wave funetion obeys the Pauli prineiple but is not an eigenfunction of the total spin operator and is thus a mixture of different spin multiplicities. In the present two-eleetron case, an alternative form of the wave funetion which has the same total energy, which is a pure singlet state, but whieh is no longer antisymmetric as required by thePauli principle, is ... 

The theoretical results provided by the large basis sets II-V are much smaller than those from previous references [15-18] the present findings confirm that the second-hyperpolarizability is largely affected by the basis set characteristics. It is very difficult to assess the accuracy of a given CHF calculation of 2(ap iS, and it may well happen that smaller basis sets provide theoretical values of apparently better quality. Whereas the diagonal eomponents of the eleetrie dipole polarizability are quadratic properties for which the Hartree-Fock limit can be estimated with relative accuracy a posteriori, e.g., via extended calculations [38], it does not seem possible to establish a variational principle for, and/or upper and lower bounds to, either and atris-... 

Now that we have decided on the form of the wave function the next step is to use the variational principle in order to find the best Slater determinant, i. e., that one particular Osd which yields the lowest energy. The only flexibility in a Slater determinant is provided by the spin orbitals. In the Hartree-Fock approach the spin orbitals (Xi 1 are now varied under the constraint that they remain orthonormal such that the energy obtained from the corresponding Slater determinant is minimal... 

This can only be true if p2 (xj, Xj) = 0. In other words, this result tells us that the probability of finding two electrons with the same spin at the same point in space is exactly zero. Hence, electrons of like spin do not move independently from each other. It is important to realize that this kind of correlation is in no way connected to the charge of the electrons but is a direct consequence of the Pauli principle. It applies equally well to neutral fermions and - also this is very important to keep in mind - does not hold if the two electrons have different spin. This effect is known as exchange or Fermi correlation. As we will show below, this kind of correlation is included in the Hartree-Fock approach due to the antisymmetry of a Slater determinant and therefore has nothing to do with the correlation energy E discussed in the previous chapter. 

The only term for which no explicit form can be given, i. e the big unknown, is of course Exc- Similarly to what we have done within the Hartree-Fock approximation, we now apply the variational principle and ask what condition must the orbitals cp fulfill in order to minimize this energy expression under the usual constraint of ((p I (pj) = ,j The resulting equations are (for a detailed derivation see Parr and Yang, 1989) ... 

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