N-electron wave function

In die HMO approximation, the n-electron wave function is expressed as a linear combination of the atomic orbitals (for the case in which the plane of the molecule coincides with the x-y plane). Minimizing the total rt-electron energy with respect to the coefficients leads to a series of equations from which the atomic coefficients can be extracted. Although the mathematical operations involved in solving the equation are not... 

Coming back to the variational principle, the strategy for finding the ground state energy and wave function should be clear by now we need to minimize the functional E[ F] by searching through all acceptable N-electron wave functions. Acceptable means in this context that the trial functions must fulfill certain requirements which ensure that these func-... 

As discussed above, it is impossible to solve equation (1-13) by searching through all acceptable N-electron wave functions. We need to define a suitable subset, which offers a physically reasonable approximation to the exact wave function without being unmanageable in practice. In the Hartree-Fock scheme the simplest, yet physically sound approximation to the complicated many-electron wave function is utilized. It consists of approximating the N-electron wave function by an antisymmetrized product4 of N one-electron wave functions (x ). This product is usually referred to as a Slater determinant, OSD ... 

In this section we will approach the question which is at the very heart of density functional theory can we possibly replace the complicated N-electron wave function with its dependence on 3N spatial plus N spin variables by a simpler quantity, such as the electron density After using plausibility arguments to demonstrate that this seems to be a sensible thing to do, we introduce two early realizations of this idea, the Thomas-Fermi model and Slater s approximation of Hartree-Fock exchange defining the X(/ method. The discussion in this chapter will prepare us for the next steps, where we will encounter physically sound reasons why the density is really all we need. 

In words, we search over all allowed, antisymmetric N-electron wave functions and the one that yields the lowest expectation value of the Hamilton operator (i. e. the energy) is the ground state wave function. 

To understand how Kohn and Sham tackled this problem, we go back to the discussion of the Hartree-Fock scheme in Chapter 1. There, our wave function was a single Slater determinant SD constructed from N spin orbitals. While the Slater determinant enters the HF method as the approximation to the true N-electron wave function, we showed in Section 1.3 that 4>sd can also be looked upon as the exact wave function of a fictitious system of N non-interacting electrons (that is electrons which behave as uncharged fermions and therefore do not interact with each other via Coulomb repulsion), moving in the elfective potential VHF. For this type of wave function the kinetic energy can be exactly expressed as... 

The primary challenge in quantum chemistry is to find a good approximation to the electronic wave function of a quantum state. We can express any N-electron wave function in a complete basis of Slater determinants, through the FCI expansion,... 

In the adiabatic approximation the N-electron wave function is assumed in the form of a single product ... 

Unfortunately, in contrast to the Cl method, an extension of the SR CC theory to the MR case is far from being straightforward, since there is no unique way in which to generalize the SR exponential Ansatz for the exact N-electron wave function jT), i.e.,... 

Approximations to an exact N electron wave function are expressed in terms of vectors in the Fock subspace F(m,N). 

From a purist theoretical point of view, there is one further important result hidden in the Levy constrained-search strategy it provides a unique, albeit only formal, route to extract the ground state wave function F, from the ground state density p0. This is anything but a trivial problem, since there are many antisymmetric N-electron wave functions that yield... 

The N-electron wave-function of a polyatomic molecule containing the atoms A, B. n is usually most conveniently expressed as a Slater product of molecular spin orbitals each of which is a linear combination of atomic orbitals multiplied by a spin function... 

The term "ab-initio" is often taken to include those methods which, given an initial choice for the general form of the N-electron wave function, (e.g, one or many determinants) attempt to solve the Schrodinger Equation... 

In the standard SR CC approach, the exact (nonrelativistic) N-electron wave function I1 ) for the state of interest (assumed to be energetically the lowest state of a given symmetry species) is represented by the so-called cluster expansion relative to some IPM wave function 4>0). This expansion is concisely expressed via the exponential cluster ansatz... 

Again there are two ways to arrive at pair functions. One can either start from a given n-electron wave function and construct pair functions from it, or one can formulate an ansatz for the wave function in terms of pair functions and obtain these by a variational or quasi-variational procedure. 

Blatt [36], Coleman [37, 38], then Bratoz and Durand [39] investigated a special N-electron wave function in which all geminals were constrained to have the same form, and established the relationship between this function and that used by Bardeen, Cooper and Shieffer (BCS) [40] to describe superconducting systems with electron pairs. The underlying wave function was termed as the antisymmetrized geminal power (AGP) function. 

It has been known for a long time, especially from the work of Dirac [15] and of Lowdin [78], that the (now idempotent) 1 DM is sufficient to determine the N-electron wave function for the case of a single Slater determinant. It has been equally clear to many workers in the field that such knowledge of the 1DM cannot be adequate to reconstruct the AT-body wave function for the fully interacting electron system, without appeal to the total Hamiltonian. 

The recent discovery of ceramic high-Tc superconductors has forced a re-examination of the basic concepts and physical assumptions employed in current theoretical approaches. In reexamining basic concepts, it is well to remember that the true N-electron wave function may be expanded in terms of components each of which is made up of N single particle functions and that this expansion can be made in (at least) two different ways ... 

A Slater determinant gives an exact representation of the n-electron wave function only in the (fictitious) limit of no interactions among the electrons (i.e., for a system of electrons described by the Hamiltonian in Eq. (4)). For a real system of interacting electrons, described by the Hamiltonian in Eq. (3), the Slater determinant can only serve as an approximate wave function. Nevertheless, in this case, we may still represent the true -electron wave function T exactly as a linear combination of Slater determinants (Eq. (13)) ... 

Having established the basis of one-electron functions, we construct a basis set for our N-electron wave functions as the set of Slater determinants that we can construct... 

This suggests that in the particle-hole representation each occupied one-particle state in the lN configuration can be assigned a value of the z-projection of the quasispin angular momentum 1/4 and each unoccupied (hole) state —1/4. When acting on an N-electron wave function the operator alls) produces an electron and, simultaneously, annihilates a hole. Therefore, the projection of the quasispin angular momentum of the wave function on the z-axis increases by 1/2 when the number of electrons increases by unity. Likewise, the annihilation operator reduces this projection by 1/2. Accordingly, the electron creation and annihilation operators must possess some tensorial properties in quasispin space. Examination of the commutation relations between quasispin operators, and creation and annihilation operators... 

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