Wavefunctions many-body

For this simple Hamiltonian, let us write the many-body wavefunction as... 

Using the orbitals, ct)(r), from a solution of equation Al.3.11, the Hartree many-body wavefunction can be constructed and the total energy detemiined from equation Al.3,3. 

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. 

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 ... 

To ensure this, the-many-body wavefunction can be written as a Slater determinant of one particle wavefunctions - this is the Hartree Fock method. The drawbacks of this method are that it is computationally demanding and does not include the many-body correlation effects. 

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 ... 

Our numerical implementation of the above scheme works as follows. We start off with an initial guesses kg for the many-body wavefunction and a corresponding guess for Vx. A fixed number Nc of statistically independent configurations Rt- are then sampled from Jq 2 and the Monte Carlo estimator of fi2 over these configurations is evaluated... 

The quality of a variational quantum Monte Carlo calculation is determined by the choice of the many-body wavefunction. The many-body wavefunction we use is of the parameterized Slater-Jastrow type which has been shown to yield accurate results both for the homogeneous electron gas and for solid silicon (14) (In the case of silicon, for example, 85% of the fixed-node diffusion Monte Carlo correlation energy is recovered). At a given coupling A, 4>A is written as... 

We performed adiabatic connection calculations for cosine-wave jellium using six values of A 0,0.2,0.4,0.6,0.8,1. The many-body wavefunctions for A > 0 were optimized by fixed-density variance minimization using 10000 independent N—electron configurations at each A. These configurations were regenerated several times. The weight factor in expression (27) was set equal... 

Another closely related constraint is that of Galileian invariance. Suppose that a many-body wavefunction (r, t2, r ) satisfies the time-independent interacting N-particle Schrddinger equation with an external one-particle potential r(r). Then, provided that the inter-particle interaction depends on coordinate differences only, it is readily verified that a boosted wavefunction of the form... 

A more precise description for this class is full wavefunction methods, where the basic variable is the full many-body wavefunction. The main problem with full wavefunction approaches is that the computational load increases drastically with the number of electrons N. At the Hartree-Fock level, the load increases as and the scaling with N increases steadily, the more... 

Because DFT-based techniques have the electronic density p r) as the basic variable, the computational load scales moderately with the number of electrons N, 0 N ). Thus they are able to handle significantly more atoms than traditional quantum chemical approaches that retain the full electronic many-body wavefunction as the basic variational quantity. This favorable scaling currently makes DFT-based techniques most promising for ab initio studies of M/C interfaces, and therefore we will emphasize this group of methods in our review. 

In the realm of quantum chemistry, the key to understanding molecular stmcture has traditionally been assumed to be the electronic wavefunction. In Hartree-Fock theory, for instance, the many-body wavefunction is taken to be a determinant formed from a set of one-electron wavefunctions. Within this framework, the exchange interaction is derived exactly, and the concepts of Pauli exclusion and electronic spin are included in a natural manner As the system size increases, however, dynamic correlations between electrons gradually become more important, and it is necessary to go beyond the Hartree-Fock theory if reasonable results are to be obtained. 

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