Wave functions exact, expansion

We begin by considering, in Section 5.1, the relationship between the expansions made in the one-particle space (the orbital space) and in the N-particle space (the Fock space). For an exact representation of the true wave function, both expansions must be complete. The one-particle expansions - which are common to all models - are not considered in detail in this chapter but are examined separately in Chapters 6-8. In the present chapter, we concentrate on the N-particle treatment, which is studied here in isolation from the one-particle treatment, proceeding on the assumption that the one-particle basis is sufficiently flexible to ensure a reasonably accurate representation of the true wave function. 

Coupled cluster calculations are similar to conhguration interaction calculations in that the wave function is a linear combination of many determinants. However, the means for choosing the determinants in a coupled cluster calculation is more complex than the choice of determinants in a Cl. Like Cl, there are various orders of the CC expansion, called CCSD, CCSDT, and so on. A calculation denoted CCSD(T) is one in which the triple excitations are included perturbatively rather than exactly. 

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 ... 

Without introducing any approximations, the total (exact) wave function can be written as an expansion in the complete set of electronic functions, with the expansion coefficients being functions of the nuclear coordinates. 

In other words, the exact wave function behaves asymptotically as a constant 4- l/2ri2 when ri2 is small. It would therefore seem natural that the interelectronic distance would be a necessary variable for describing electron correlation. For two-electron systems, extremely accurate wave functions may be generated by taking a trial wave function consisting of an orbital product times an expansion in electron coordinates such as... 

Ho is the normal electronic Hamilton operator, and the perturbations are described by the operators Pi and P2, with A determining the strength. Based on an expansion in exact wave functions, Rayleigh-Schrddinger perturbation theory (section 4.8) gives the first- and second-order energy collections. 

To improve on the wave function one has to accept that the standard multideterminantal expansion [Eq. (13.3)] is unsuitable for near-exact but practical approximations to the electronic wavefunction. The problem is dear from a simple analysis of the electronic Hamiltonian in Eq. (13.2) singularities in the Coulomb potential at the electron coalescence points necessarily lead to irregularities in first and higher derivatives of the exact wave function with respect to the interpartide coordinate, rj 2. The mathematical consequences of Coulomb singularities are known as electron-electron (correlation) and electron-nuclear cusp conditions and were derived by... 

Establishing a hierarchy of rapidly converging, generally applicable, systematic approximations of exact electronic wave functions is the holy grail of electronic structure theory [1]. The basis of these approximations is the Hartree-Fock (HF) method, which defines a simple noncorrelated reference wave function consisting of a single Slater determinant (an antisymmetrized product of orbitals). To introduce electron correlation into the description, the wave function is expanded as a combination of the reference and excited Slater determinants obtained by promotion of one, two, or more electrons into vacant virtual orbitals. The approximate wave functions thus defined are characterized by the manner of the expansion (linear, nonlinear), the maximum excitation rank, and by the size of one-electron basis used to represent the orbitals. 

At this place it should be mentioned that the BW expansion (8) at the same time prescribes the normalization of the exact wave functions, i.e. 

One could insist that the mass dependence in (5.9) is natural because the calculations leading to it are done without expansion over the mass ratio, and are therefore exact. On the other hand, the factor m / mM) symmetric with respect to the masses naturally arises in all apparently nonrecoil calculations just from the Schrodinger-Coulomb wave function squared. According to the tradition we preserve the coefficient before the logarithm squared term in the form given in (3.53). Then the new contribution contained in (5.9) has the form [12, 13]... 

Most of the formalism to be developed in the coming sections of these lecture notes will be independent of the specific definition of the configurational basis, in which we expand the wave function. We therefore do not have to be very explicit about the exact nature of the basis states hn>. They can be either Slater determinants or spin-adapted Configuration State Functions (CSF s). For a long time it was assumed that CSF s were to be preferred for MCSCF calculations, since it gives a much shorter Cl expansion. Efficient methods like GUGA had also been developed for the solution of the Cl problem. Recent... 

If model function xpo is known, then we can find a series expansion for the remaining part Qxp of the exact wave function (3.8). Starting with the Schrodinger equation in the form... 

Eq. (3.12) represents the Brillouin-Wigner (BW) perturbation expansion of the exact wave function, whereas the corresponding expression for the... 

The relation between the CFP with a detached electrons and the reduced matrix elements of operator q>(lNfiLS generating [see (15.4)] the <7-electron wave function is established in exactly the same way as in the derivation of (15.21). Only now in the appropriate determinants we have to apply the Laplace expansion in terms of a rows. The final expression takes the form... 

This approach was developed originally as an approximate method, if the wave functions of isolated atoms are taken as a basis wave functions Wannier functions. Only in the last case the expansion (1) and the Hamiltonian (2) are exact, but some extension to the arbitrary basis functions is possible. In principle, the TB model is reasonable only when local states can be orthogonalized. The method is useful to calculate the conductance of complex quantum systems in combination with ab initio methods. It is particular important to describe small molecules, when the atomic orbitals form the basis. 

In this section we shall discuss an approach which is neither variational nor perturba-tional. This approach also has its origin in nuclear physics and was introduced to quantum chemistry by Sinanoglu47, It is based on a cluster expansion of the wave function. A systematic method for the calculation of cluster expansion components of the exact wave function was developed by C ek48 The characteristic feature of this approach is the expansion of the wave function as a linear combination of Slater determinants. Formally, this expansion is similar to the ordinary Cl expansion. The cluster expansion, however, gives us not only the physical insight of the correlation energy but it also shows the connections between the variational approaches (Cl) and the perturbational approaches (e.g. MB-RSPT). 

The Rayleigh-Schrodinger perturbation expansion for the exact wave function to the first order is given by... 

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