Many-body electronic wavefunction

Two commonly used approximations are the Hartree-Fock approach and density-functional theory (DFT). The Hartree-Fock approach approximates the exact solution of the Schrodinger equation using a series of equations that describe the wavefunc-tions of each individual electron. If these equations are solved explicitly during the calculation, the method is known as ab initio Hartree-Fock. The less expensive (i.e., less time-consuming) semi-empirical methods use preselected parameters for some of the integrals. DFT, on the other hand, uses the electronic density as the basic quantity, instead of a many-body electronic wavefunction. The advantage of this is that the density is a function of only three variables (instead of 3N variables), and is simpler to deal with both in concept and in practice. 

An alternative to computing the properties of many-body electronic systems via the wavefunction is computing these properties via the electronic density. This approach was invented by Hohenberg and Kohn, who showed that the total energy of the many-body electronic system can be expressed as a functional of the density in the following way ... 

There are numerous approaches to approximate solutions for (1), most of which involve finding the system s total ground state energy, E, including methods that treat the many-body wavefunction as an antisymmetric function of one-body orbitals (discussed in later sections), or methods that allow a direct representatiOTi of many-body effects in the wave function such as Quantum Monte Carlo (QMC), or hybrid methods such as coupled cluster (CC), which adds multi-electrOTi wave-function corrections to account for the many-body (electron) correlatirais. 

Once a complex valued electronic density matrix is in hand, it is not difficult at all to calculate the electron flux and it is not difficult either to transform the flux information to photoelectron signals (from body frame to laboratory frame), as successfully accomplished by Bandrauk et al. [499, 500], On the other hand, it is not as simple to track the simultaneous dynamics of remaining electrons in the molecular site while photoelectrons are leaving away. Note that even if we successfully have a rescaled density p (r, t) in equation (7.2) with an estimate of the flux, we can make infinitely many (untrue) electronic wavefunctions that can reproduce this same p r,t). The key to be considered is, therefore, to assign which electronic configurations the photoelectrons are leaving from and how consistently we can describe the states of the remaining electronic wavepacket. To do so, we think it to be most appropriate to use natural orbital representation of the electron flux. 

Applying Flartree-Fock wavefiinctions to condensed matter systems is not routine. The resulting Flartree-Fock equations are usually too complex to be solved for extended systems. It has been argried drat many-body wavefunction approaches to the condensed matter or large molecular systems do not represent a reasonable approach to the electronic structure problem of extended systems. 

In our hydrogen molecule calculation in Section 2.4.1 the molecular orbitals were provided as input, but in most electronic structure calculations we are usually trying to calculate the molecular orbitals. How do we go about this We must remember that for many-body problems there is no correct solution we therefore require some means to decide whether one proposed wavefunction is better than another. Fortunately, the variation theorem provides us with a mechanism for answering this question. The theorem states that the... 

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. 

The interactions between electrons are inherently many-body forces. There are several methods in common use today which try to incorporate some, or all, of the many-body quantum mechanical effects. An important term is that of electronic exchange [57, 58]. Mathematically, when two particles in the many-body wavefunction are exchanged the wavefunction changes sign ... 

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. 

All three forms of the dipole matrix element are equivalent because they can be transformed into each other. However, this equivalence is valid only for exact initial- and final-state wavefunctions. Since the Coulomb interaction between the electrons is responsible for many-body effects (except in the hydrogen atom), and the many-body problem can only be solved approximately, the three different forms of the matrix element will, in general, yield different results. The reason for this can be seen by comparing for the individual matrix elements how the transition operator weights the radial parts R r) and Rf(r) of the single-particle wavefunction differently ... 

If we are interested in the ground-state electronic properties of a molecule or solid with a given set of nuclear coordinates we should seek the solution to the Schrodinger equation which corresponds to the lowest electronic energy of the system. However, the inter-electronic interactions in eq. (2.2) are such that this differential equation is non-separable. It is therefore impossible to obtain the exact solution to the full many-body problem. In order to proceed, it is necessary to introduce approximation in this equation. Two types of approximations can be separated, namely, approximations of the wavefunction, VF, from a true many-particle wavefunction to, in most... 

Modern density functional theory (DFT) provides an enormous simplification of the many-body problem [1-7], It enables one to replace the complicated conventional wavefiinction approach with the simpler density functional formalism. The ground-state properties of the system under investigation are obtained through a minimization over densities rather than a minimization over wavefunctions. The electron density is especially attractive for calculations involving large systems, because it contains only three dimensions regardless of the size of the system. In addition, even for relatively small systems, it has been found that density-functional methods, for certain situations, often yield results competitive with, or superior to those obtained from various traditional wavefiinction approaches. 

We consider an N-electron system having ground-state density n(r). At a given coupling constant A the corresponding many-body wavefunction satisfies... 

The above unconstrained optimization cannot be directly applied at intermediate values of A for which the Hamiltonian contains the unknown potential V. We found, however, that a simultaneous determination of and V can be achieved by performing the following constrained optimization. We assume that the trial many-body wavefunction results in the electron density rcA(r) and expand both nA(r) and the ground state density n(r) in a complete and orthonormal set of basis functions fs ... 

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