Many-electron wavefunctions

What happens when we have a many-electron wavefunction, such as the one below which relates to the simple valence-bond treatment of dihydrogen ... 

Integration of P(r) with respect to the coordinates of this electron (now written r) gives the number of electrons, 2 in this case. In the case of a many-electron wavefunction that depends on the spatial coordinates of electrons 1,2,..., m, we define the electron density as... 

Once electron repulsion is taken into account, this separation of a many-electron wavefunction into a product of one-electron wavefunctions (orbitals) is no longer possible. This is not a failing of quanmm mechanics scientists and engineers reach similar conclusions whenever they have to deal with problems involving three or more mutually interacting particles. We speak of the three-body problem. 

There are m doubly occupied molecular orbitals, and the number of electrons is 2m because we have allocated an a and a spin electron to each. In the original Hartree model, the many-electron wavefunction was written as a straightforward product of one-electron orbitals i/p, i/ and so on... 

Note that, throughout this discussion, we have used lower-case letters when refering to orbitals and upper-case when we mean many-electron wavefunctions. There arises the question of, what are the relationships between I and L, or between s and S . They are determined by the vector coupling rule. This states that the angular momentum for a coupled (i.e. interacting) pair of electrons may take values ranging from their sum to their difference (Eq. 3.11). 

Term wavefunctions describe the behaviour of several electrons in a free ion coupled together by the electrostatic Coulomb interactions. The angular parts of term wavefunctions are determined by the theory of angular momentum as are the angular parts of one-electron wavefunctions. In particular, the angular distributions of the electron densities of many-electron wavefunctions are intimately related to those for orbitals with the same orbital angular momentum quantum number that is. 

The identity of (b) with (a) is obvious. In (c), a section of the density is shown to take the form of a spherical density from which a density of the form (b) has been subtracted. Alternatively, (c) may be viewed as a distribution of positive charge in the form of (b). Whether components of a D term take the form (b) or (c) depends upon the number of electrons described by the many-electron wavefunction. 

In an octahedral crystal field, for example, these electron densities acquire different energies in exactly the same way as do those of the J-orbital densities. We find, therefore, that a free-ion D term splits into T2, and Eg terms in an octahedral environment. The symbols T2, and Eg have the same meanings as t2g and eg, discussed in Section 3.2, except that we use upper-case letters to indicate that, like their parent free-ion D term, they are generally many-electron wavefunctions. Of course we must remember that a term is properly described by both orbital- and spin-quantum numbers. So we more properly conclude that a free-ion term splits into -I- T 2gin octahedral symmetry. Notice that the crystal-field splitting has no effect upon the spin-degeneracy. This is because the crystal field is defined completely by its ordinary (x, y, z) spatial functionality the crystal field has no spin properties. 

Approximations have been reviewed in the case of short deBroglie wavelengths for the nuclei to derive coupled quantal-semiclassical computational procedures, by choosing different types of many-electron wavefunctions. Time-dependent Hartree-Fock and time-dependent multiconfiguration Hartree-Fock formulations are possible, and lead to the Eik/TDHF and Eik/TDMCHF approximations, respectively. More generally, these can be considered special cases of an Eik/TDDM approach, in terms of a general density matrix for many-electron systems. 

The Pauli antisymmetry principle is a requirement a many-electron wavefunction must obey. A many-electron wavefunction must be antisymmetric (i.e. changes sign) to the interchange of the spatial and spin coordinates of any pair of electrons i and/, that is ... 

Unfortunately, the many-electron wavefunction itself does not necessarily provide insight into the chemistry of complex molecules as it describes the electronic distribution over the whole system. It is therefore assumed that the true many-electron wavefunction. can be represented as the product of a series of independent, one-electron wavefunctions ... 

Applying the permutation operator P12 is therefore equivalent to interchanging rows of the determinant in Eq. (2.15). Having devised a method for constructing many-electron wavefunctions as a product of MOs, the final problem concerns the form of the many-electron Hamiltonian which contains terms describing the interaction of a given electron with (a) the fixed atomic nuclei and (b) the remaining (N— 1) electrons. The first step is therefore to decompose H(l, 2, 3,..., N) into a sum of operators Hj and H2, where ... 

The energy associated with the many-electron wavefunction is then given by ... 

The final equation for the energy of a given many-electron wavefunction in terms of its spatial MOs [Eq. (2.18)] can then be written as ... 

This integral is that of a core-electron interaction and therefore available through solution of the many-electron wavefunction using a variety of methods. 

Although in many cases, particularly in PE spectroscopy, single configurations or Slater determinants 2d> (M+ ) were shown to yield heuristically useful descriptions of the corresponding spectroscopic states 2 f i(M+ ), this is not generally true because the independent particle approximation (which implies that a many-electron wavefunction can be approximated by a single product of one-electron wavefunctions, i.e. MOs 4>, as represented by a Slater determinant 2 j) may break down in some cases. As this becomes particularly evident in polyene radical cations, it seems appropriate to briefly elaborate on methods which allow one to overcome the limitations of single-determinant models. 

In analogy to using a linear combination of atomic orbitals to form MOs, a variational procedure is used to construct many-electron wavefunctions from a set of N Slater determinants y, i.e. one sets up a N x. N matrix of elements flij = (d>, H d>y) which, upon diagonalization, yields state energies and associated vectors of coefficients a used to define (fi as a linear combination of A,s ... 

However, if this is not the case, the perturbations are large and perturbation theory is no longer appropriate. In other words, perturbation methods based on single-determinant wavefunctions cannot be used to recover non-dynamic correlation effects in cases where more than one configuration is needed to obtain a reasonable approximation to the true many-electron wavefunction. This represents a serious impediment to the calculation of well-correlated wavefunctions for excited states which is only possible by means of cumbersome and computationally expensive multi-reference Cl methods. 

As was mentioned previously, simple orbital products (electron configurations) must be converted into antisymmetrized orbital products (Slater determinants) in order to satisfy the Pauli principle. Thus, proper many-electron wavefunctions satisfy constraints of exchange antisymmetry that have no counterpart in pre-quantum theories. 

The basic problem is to solve the time-independent electronic Schrodinger equation. Since the mass of the electrons is so small compared to that of the nuclei, the dynamics of nuclei and electrons can normally be decoupled, and so in the Born-Oppenheimer approximation the many-electron wavefunction P and corresponding energy may be obtained by solving the time-independent Schrodinger equation in which the nuclear positions are fixed. We thus solve... 

The many-electron wavefunction must obey the Pauli Principle, i.e. possess the right permutation symmetry, such that it changes sign when any two electrons are exchanged, i.e. 

The most usual starting point for approximate solutions to the electronic Schrodinger equation is to make the orbital approximation. In Hartree-Fock (HF) theory the many-electron wavefunction is taken to be the antisymmetrized product of one-electron wavefunctions (spin-orbitals) ... 

We consider an /V-electron system where the electrons experience the mutual Coulomb interaction along with an external potential vv( r) due to the nuclei. The system is then subjected to an additional TD scalar potential (r, t) and a TD vector potential A(r, t). The many-electron wavefunction [Pg.74]

Since the time that Coulson [7] discussed the promise and challenges of computing the energies and properties of atoms and molecules without the many-electron wavefunction, quantum chemistry has experienced many important advances toward the accurate treatment of electron correlation including the... 

Importantly, the anti-Hermitian CSE may be evaluated through second order of a renormalized perturbation theory even when the cumulant 3-RDM is neglected in the reconstruction. The anti-Hermitian part of the CSE [27, 31, 63] is the stationary condition for two-body unitary transformations of the A-particle wave-function [31, 32], and hence the two-body unitary transformations may easily be evaluated with the anti-Hermitian CSE and RDM reconstruction without the many-electron Schrodinger equation. The contracted Schrodinger equation in conjunction with the concepts of reconstruction and purification provides a new, important approach to computing the 2-RDM directly without the many-electron wavefunction. 

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