Variance of the local energy

For example, the variance of the local energy, E oc = W/V , has long been use to gauge the goodness of an approximate wave function, The variance, will approach zero as the true wave function is approached, since then the local energy will be a constant equal to the exact energy. Note that the... 

H Pt E t, the variance of the local energy approaches zero as approaches the exact eigenfunction. The relatively small variance of the local energy allows VMC to estimate energies much more efficiently than other properties whose operators do not commute with the Hamiltonian. VMC can also be used to evaluate more general matrix elements of the form Aij = ( I, This integral may be rewritten in... 

The factors that contribute to Nmc have been examined previously by Ceperley [159]. First, the variance of the local energy increases with molecular size, so the number of independent MC points must increase to reduce the error to the same tolerance. Second, the MC points will not be independent of each other until the random walk has been given sufficient time for the walker positions to decorrelate. Third, the time step bias of a DMC calculation increases with system size, so a larger number of steps must be taken before the decorrelation time is achieved. 

This condition indicates that the variance of the local energy (R) is zero if and only if the function 4 t is a true eigenfunction of the Hamiltonian H ... 

This is the variational principle for the variance of the local energy. The evaluation of the variance is easy with the MC method. Thus, the variance is often adopted as the objective function in the optimization of the wave function within the QMC method. The efficient optimization of the wave function is an important problem in the QMC method and several works can be found in this line [23-30]. 

The energy is obtained as a weighted average of the local energy over the sample. The efficiency of the calculation is therefore determined by the time required to calculate E, the fluctuations of the weights, and the variance of the local energy = (El) - Eff. 

The Slater determinants are often taken from HF, DFT or MC-SCF/ CASSCF calculations. The Jastrow correlation function is often parametrized in the Schmidt-Moskowitz form where U is expanded in powers of the scaled distances r, and fy,f = r/ + ar). This expansion goes back to Boys and Handy. Other expansions such as Pade-type or expansions in terms of unsealed distances with cutoff parameters have been used successfully. The Jastrow factor does not have to be rotationally invariant. Riley and Anderson demonstrated that a directional Jastrow term can improve the VMC energy of LiH considerably. The parameter vector p of U x p) can be determined efficiently by minimizing the variance of the local energy in the form... 

Simple correlation functions of the form found in equation (6) were employed in early QMC computations. These functions were quite useful because the electron-electron cusp conditions were easily satisfied, giving a large reduction in the variance of the local energy. An improved correlation function for atoms was introduced by Unuigar et al. in their treatment of atoms with 2, 4, and 10 electrons. This correlation function is of the form... 

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