Quantum Monte Carlo for Atoms, Molecules and Solids
Download 242 Kb.
|
- Navigate this page:
- Trial wave function optimization
- Evaluation of the trial function: an approach to Linear Scaling
Auxiliary Field QMCRecent studies by Zhang and collaborators show favorable results using a real phase form of AFQMC. In this approach, …[with refs.] Trial Wave FunctionsForm of the Trial FunctionThe types of wave functions that have been used in VMC and DMC begin with those of basis set quantum chemistry, referring primarily to Hartree-Fock (HF) and various multi-configuration formalisms. The latter include multi-configuration self-consistent-field (MCSCF) and configuration interaction (CI). In addition, various density functional theory (DFT) wave functions have been used, in particular, a number of generalized gradient forms [refs.]. There has been some use of Moeller-Plesset perturbation theory functions, but state-of-the-art coupled cluster forms have not been used as trial wave functions owing to the large number of operations required to evaluate such functions the numerous times required by QMC. The trial wave functions play two important roles in the QMC calculations. The first is to avoid the fermion sign problem by use of the fixed-node approximation. The second is efficiency: the statistical fluctuations are significantly reduced by increasing trial function accuracy. This is practically very important since the efficiency gain can reach two to three orders of magnitude. For cutting-edge applications that involve large numbers of electrons or high statistical accuracy, trial function quality is crucial and determines whether such can calculations are feasible. To satisfy both these requirements trial functions one chooses the following general form:
where  is an antisymmetric wave function and the second term is a symmetric factor which includes explicit electron-electron correlations, i.e., terms explicitly dependent on interparticle coordinates. Focusing first on the correlation function one can expand  in functions of two- and three- (an so forth) particle interactions,
where  describes correlations between electrons i and j, and  describes the dominant 3-particle correlations of the type electron-electron-ion where I labels ions – to date higher order correlations have not be explored. The correlation functions  and  are further expanded in appropriate basis sets with expansion coefficients chosen to satisfy know analytic behavior of the exact wave function (typically the two-particle cusp conditions, satisfaction of which will keep the local energy bounded when two particles meet) and then optimized by VMC methods. A simple form of correlation function is the Pade-Jastrow (often referred to as the Jastrow factor) form
Note that by including explicit correlation into the trial wave function we are able recover a large fraction of the correlation energy at the VMC level and significantly reduce the variance of the local energy by satisfying the electron-electron cusp conditions. In recent years, considerable effort has been invested in improvements of the antisymmetric function,
or, more generally, a linear combination of such products,
A single determinantal product multiplied by a Jastrow factor is often called a Slater-Jastrow wave function. The one-electron orbitals from various methods have been tested and utilized in such wave functions. Most popular have been Hartree-Fock (HF) and post-HF orbitals such as those from multi-configuration self-consistent-field (MCSCF) and configuration interaction (CI) approaches. Orbitals from density functional theory (DFT) have also been employed, in particular, a number of generalized gradient and hybrid (with mixed exact Fock exchange) have been explored [Wagner1,Wagner2]. In addition, there has been some use of orbitals generated from Moeller-Plesset perturbation theory, as well as direct optimization of the one-electron orbitals together with other variational parameters as will be mentioned below. Most recently, pair-orbitals (geminals) have been introduced into QMC. This direction was pioneered in the work of Sorella’s group [Casula1, Casula2] by employing the antisymmetrized product of pairing orbitals which is known in condensed matter physics as Bardeen-Cooper-Schrieffer (BCS) wave function, given as,
where  is the singlet pair orbital assuming that the total spin of the system is zero. Although the form can be generalized to include spin polarized states, for the fully spin-polarized state the wave function becomes uncorrelated like in HF. A more general form has been employed by Bajdich and coworkers [Bajdich1, Bajdich2]
which has a form of a pfaffian [ref] of the pair spin-orbitals. By assigning electron spins one can write the form as a pfaffian with pair singlet, triplet and unpaired spatial orbitals and thus describe correlation of any spin state on the same systematic pairing level as given by
Psi_PF = pf [matrix of chi^^, phi^v,chivv etc equation (5) in Bajdich1] The pair orbitals are expanded in an appropriate one-particle basis as given by
(Brian please retype the expressions for phi and chi as given below) phi(i,j) = sum k>=l c_kl[gamma_k (i) gamma_l (j) + gamma_l(i)gamma_k(j)] chi(i,j)= sum k>l c_kl[gamma_k (i) gamma_l (j) – gamma_l(i)gamma_k(j)] For calculations with a sizable number of electrons it is important that the pfaffian can be evaluated by algorithms which are analogous to Gaussian elimination for determinants and have the same computational scaling. This proved to be quite successful and single pfaffians lead to systematically improved results for the recovered correlation energy as has been tested on the first row atoms and molecules. Further improvements include expansion in pfaffians which are much more compact than expansions in determinants typically decreasing the number of terms by an order of magnitude while recovering 98-99 % of the correlation energy for the first row systems [Bajdich 2]. Further research in this direction is in progress. Less traditional forms : transcorrelated constructions etc.
Trial wave function optimizationOptimization of trial wave functions in QMC is a complex and challenging task due to the overall nonlinearity of the optimization and stochastic integral evaluation. Any optimization algorithm used must deal with multiple issues of the statistical "noise" of the quantities being optimized (typically the local energy or its variance), potential linear dependencies in the parameters since they often constitute an over complete set, and the problems of local minima. Many of these problems have been effectively resolved in recent years and currently tens or hundreds parameters can be optimized with good efficiency [Umrigar1, Umrigar2, Rappe]. These developments have enabled the concurrent optimization of Jastrow parameters, determinantal expansion coefficients, and one-particle orbitals for the C2 molecule leading to an accuracy of 0.XXXX eV, { equal or better than experiment?}. Very recently, of the order of ten of thousands parameters have been effectively optimized in simulations of liquid hydrogen [Sorella1]. Evaluation of the trial function: an approach to Linear ScalingOne of the most computationally intensive steps in the QMC algorithm is the evaluation of trial function, and specifically the Slater determinant. For increasingly larger molecules, the number of operations needed to evaluate the determinant and hence the energy at each point of a QMC simulation increases  where N is the rank of the determinant. For large systems the determinant becomes increasingly sparse because the interaction between electrons diminishes exponentially with distance [Maslen], and several approaches have been introduced to take advantage of this property to reduce the scaling of the computational effort to a linear function of N.
In various implementations a number of steps have been taken in order to obtain near-linear scaling [Manten, Williamson]. In particular, wave function evaluation in the QMC code Zori follows the procedure of Aspuru-Guzik et al. [Aspuru-Guzik3] in which a sparse representation of the Slater matrix is used leading to evaluation of only the non-zero elements of the wave. This approach, combined with the use of localized orbitals in a Slater-type-orbital basis set, significantly extends the size of molecule that can be treated with the QMC method – in this case the largest system contained 390 electrons. Tests confirm that this approach leads to linear scaling of Slater determinant evaluation with system size, see [Lester2]. With this approach, the calculation of excitation energies scales as N2. Download 242 Kb. Share with your friends:
|
because it determines the fermion nodes and therefore controls the fixed-node bias. The simplest antisymmetric wave function is a Slater determinant of spin orbitals. For QMC trial functions it has been shown [ref] that this determinant can be factored into a product of spin-up and spin-down Slater determinants of one-particle orbitals,
.
]
