Characteristics of the exact wave function

The time-independent molecular electronic Schrodinger equation for an IV-electron system with the Hamiltonian H in the coordinate representation reads 

One popular strategy - which forms the basis for the methods examined in this book -is to seek an approximate solution to the Schrodinger equation (4.1.1) as a linear combination of Slater determinants constructed from a set of orthonormal orbitals  

In the limit of a complete set of orbitals, we may arrive at the exact solution to the Schrodinger equation in the form (4.1.2) but the determinantal expansion then becomes infinite. In practice, we must resort to truncated expansions and thus be satisfied with approximate solutions to the electronic SchrOdinger equation. 

Approximations should not be made in a haphazard manner. Rather we should seek to retain in our wave function as many symmetries and properties of the exact solution as possible. Indeed, some of the characteristics of the exact wave function are so important that we should try to incorporate them at each level of theory, and a few are so fundamental that they are introduced into our models without thought. We here list some of these properties and symmetries of the exact wave function that may guide us in the construction of models. We note that many of these characteristics or symmetries of the exact solution to the Schiddinger equation may be written in the form of some supplementary eigenvalue equation 

The exact state is a function of the space and spin coordinates of N electrons. The approximate state 0) should therefore be an eigenfunction of the number operator (1.3.12) with an eigenvalue equal to the number of electrons  

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The reasons for not invoking the variation principle in the optimization of the wave function are given in Chapter 13, which provides a detailed account of coupled-cluster theory. We here only note that the loss of the variational property characteristic of the exact wave function is unfortunate, but only mildly so. Thus, even though the coupled-cluster method does not provide an upper bound to the FCI energy, the energy is usually so accurate that the absence of an upper bound does not matter anyway. Also, because of the Lagrangian method of Section 4.2.8, the complications that arise in connection with the evaluation of molecular properties for the nonvariational coupled-cluster model are of little practical consequence. 

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