Hartree-Fock approximation occupied spin orbitals

In the ordinary Hartree-Fock scheme, the total wave function is approximated by a single Slater determinant and, if the system possesses certain symmetry properties, they may impose rather severe restrictions on the occupied spin orbitals see, e.g., Eq. 11.61. These restrictions may be removed and the total energy correspondingly decreased, if instead we approximate the total wave function by means of the first term in the symmetry adapted set, i.e., by the projection of a single determinant. Since in both cases,... 

Configuration interaction (Cl) is conceptually the simplest procedure for improving on the Hartree-Fock approximation. Consider the determinant formed from the n lowest-energy occupied spin orbitals this determinant is o) and represents the appropriate SCF reference state. In addition, consider the determinants formed by promoting one electron from an orbital k to an orbital v that is unoccupied in these are the singly excited determinants ). Similarly, consider doubly excited (k, v,t) determinants and so on up to n-tuply excited determinants. Then use these many-electron wavefimctions in an expansion describing the Cl many-electron wavefunction [Pg.13]

The fact that there are only two kinds of spin function (a and (1), leads to the conclusion that two electrons at most may occupy a given molecular orbital. Were a third electron to occupy the orbital, two different rows in the determinant would be the same which, according to the properties of determinants, would cause it to vanish (the value of the determinant would be zero). Thus, the notion that electrons are paired is really an artifact of the Hartree-Fock approximation. 

Note that in the Hamiltonian, the Coulombic repulsion of the electrons is spin-independent This suggests that when trying to improve the description (by going beyond the Hartree-Fock approximation), we have to worry more about correlation of electrons with the opposite spin functions (e.g., those occupying the same orbital). 

It must be remarked that in Blumberg s calculations, a restricted Hartree-Fock approximation has been used in computing the spin density at various nuclei. However, in the uiurestricted Hartree-Fock approximation where occupied orbitals with opposite spins do not have the same spatial wave function, there would be an additional contribution to the spin density at the nuclei and hence to the Knight shift. This additional contribution could also... 

The next step is to make the Hartree-Fock self-consistent field (HF-SCF) approximation as described previously for a multi-electron atom in Section 8.4. The Hartree-Fock approximation results in separation of the electron motions resulting (along with the Pauli principle) in the ordering of the electrons into the molecular orbitals as shown in Figure 9-5 for carbon monoxide. Hence, the many-electron wavefunction i for an N-electron molecule is written in terms of one-electron space wavefunctions,/, and spin functions, a or p, like what was done for complex atoms in Section 8.4. At this stage it is assumed that the N-electron molecule is a closed-shell molecule (all the electrons are paired in the occupied molecular orbitals). How molecules with open shells are represented will be discussed later in this Section. 

A Hbasis functions provides K molecular orbitals, but lUJiW of these will not be occupied by smy electrons they are the virtual spin orbitals. If u c were to add an electron to one of these virtual orbitals then this should provide a means of calculating the electron affinity of the system. Electron affinities predicted by Konpman s theorem are always positive when Hartree-Fock calculations are used, because fhe irtucil orbitals always have a positive energy. However, it is observed experimentally that many neutral molecules will accept an electron to form a stable anion and so have negative electron affinities. This can be understood if one realises that electron correlation uDiild be expected to add to the error due to the frozen orbital approximation, rather ihan to counteract it as for ionisation potentials. 

Pople refers to a specific set of approximations as defining a theoretical model. Hence the ab initio or Hartree-Fock models employ the Born-Oppenheimer, LCAO and SCF approximations. If the system under study is a closed-shell system (even number of electrons, singlet state), the constraint that each spatial orbital should contain two electrons, one with a and one with P spin, is normally made. Such wavefunctions are known as restricted Hartree-Fock (RHF). Open-shell systems are better described by unrestricted Hartree-Fock (UHF) wavefunctions, where a and P electrons occupy different spatial orbitals. We have seen that Hartree-Fock (HF) models give rather unreliable energies. 

The Hartree Fock determinant describes a situation where the electrons move independently of one another and where the probability of finding one electron at some point in space is independent of the positions of the other electrons. To introduce correlation among the electrons, we must allow the electrons to interact among one another beyond the mean field approximation. In the orbital picture, such interactions manifest themselves through virtual excitations from one set of orbitals to another. The most important class of interactions are the pairwise interactions of two electrons, resulting in the simultaneous excitations of two electrons from one pair of spin orbitals to another pair (consistent with the Pauli principle that no more than two electrons may occupy the same spatial orbital). Such virtual excitations are called double excitations. With each possible double excitation in the molecule, we associate a unique amplitude, which represents the probability of this virtual excitation happening. The final, correlated wave function is obtained by allowing all such virtual excitations to happen, in all possible combinations. 

This is the same as the number of different single determinants that one can form from N electrons and 2K spin orbitals the Hartree-Fock ground state is just one of these. A convenient way of describing these other determinants is to consider the Hartree-Fock ground state (2.64) to be a reference state and to classify other possible determinants by how they differ from the reference state, i.e., by stating which occupied or hole spin orbitals of the set Xa in (2.64), have been replaced by which virtual or particle spin orbitals of the set These other determinants can be taken to represent approximate excited states of the system or, as we shall see shortly, they can be used in linear combination with Fo> for more accurate description of the ground state or any excited state of the system. 

We have placed special emphasis on using a consistent notation throughout the book. Since quantum chemists use a number of different notations, it is appropriate to define the notation we have adopted. Spatial molecular orbitals (with latin indices ij, k...) are denoted by These are usually expanded in a set of spatial (atomic) basis functions (with greek indices ju, V, A,...) denoted by 0. Molecular spin orbitals are denoted by x Occupied molecular orbitals are specifically labeled by a, b, c,... and unoccupied (virtual) molecular orbitals are specifically labeled by r, s, r,... Many-electron operators are denoted by capital script letters (for example, the Hamiltonian is Jf), and one-electron operators are denoted by lower case latin letters (for example, the Fock operator for electron-one is /( )). The exact many-electron wave function is denoted by O, and we use T to denote approximate many-electron wave functions (i.e., the Hartree-Fock ground state wave function is o while FS is a doubly excited wave function). Exact and approximate energies are denoted by S and , respectively. All numerical quantities (energies, dipole moments, etc.) are given in atomic units. 

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