Trial wavefunctions

The ideal trial wavefunction is simple and compact, has simple easily evaluated first and second derivatives, and is accurate everywhere. Because the local energy must be evaluated repeatedly, the computation effort required for the derivatives makes up a large part of the overall computation effort for many systems. The typical trial wavefunctions of analytic variational calculations are not often useful, since they are severely restricted in form by the requirement that they be amenable to analytic integrations. The QMC functions are essentially unrestricted in form, since no analytic integrations are required. First and second derivatives of trial wavefunctions are needed, but differentiation is in general much easier than integration, and most useful trial wavefunctions have reasonably simple analytical derivatives. In most analytic variational calculations to date, it has not been possible to include the interelectron distances r,y in the trial wavefunction, and these wavefunctions are not usually explicitly correlated, whereas for QMC calculations of all types, explicitly correlated functions containing are the norm. 

A simple wavefunction for H2 in its ground electronic state may be written as 

The Jastrow term has the effect of keeping the electrons apart, thus introducing electron-electron correlation. For most molecules, even the simplest trial wave-functions at the SCF level are remarkably accurate. For hydrocarbons a singledeterminant SCF function constructed with a minimal basis set and mildly 

The typical trial wavefunction for QMC calculations on molecular systems consists of the product of a Slater determinant multiplied by a second function, which accounts to some extent for electron correlation with use of interelectron distances. The trial wavefunctions are most often taken from relatively simple analytic variational calculations, in most cases from calculations at the SCF level. Thus, for the 10-electron system methane, the trial function may be the product of the SCF function, which is a 10 x 10 determinant made up of two 5x5 determinants, and a Jastrow function for each pair of electrons. 

The values of the Jastrow constants b and c may be specified as Vi for pairs of electrons with opposite spins and as % for pairs with identical spins. This avoids infinities in the local energy for two electrons at the same position. The Jastrow functions incorporate the main effects of electron-electron interactions and give a significant improvement over simple SCF trial functions. 

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Umrigar C J, Wilson K G and Wilkins J W 1988 Optimized trial wavefunctions for quantum Monte Carlo calculations Phys. Rev. Lett. 60 1719-22... 

Variational methods, in particular the linear variational method, are the most widely used approximation techniques in quantum chemistry. To implement such a method one needs to know the Hamiltonian H whose energy levels are sought and one needs to construct a trial wavefunction in which some flexibility exists (e.g., as in the linear variational method where the Cj coefficients can be varied). In Section 6 this tool will be used to develop several of the most commonly used and powerful molecular orbital methods in chemistry. 

In all such trial wavefunctions, there are two fundamentally different kinds of parameters that need to be determined- the Cl coefficients Ci and the ECAO-MO coefficients describing the (jiik. The most commonly employed methods used to determine these parameters include ... 

To properly describe electronic rearrangement and its dependence on both nuclear positions and velocities, it is necessary to develop a time-dependent theory of the electronic dynamics in molecular systems. A very useful approximation in this regard is the time-dependent Hartree-Fock approximation (34). Its combination with the eikonal treatment has been called the Eik/TDHF approximation, and has been implemented for ion-atom collisions.(21, 35-37) Approximations can be systematically developed from time-dependent variational principles.(38-41) These can be stated for wavefunctions and lead to differential equations for time-dependent parameters present in trial wavefunctions. 

This condition is termed the variational principle. Thus, the trial wavefunction can be optimized using standard techniques43 until the system energy is minimized. At this point, the final solution can be regarded as Mf. for all practical purposes. It is clear, however, that the wavefunction that is obtained following this iterative procedure will depend on the assumptions employed in the optimization procedure. 

Given the trial wavefunction - the Slater determinant eq. (11.37) - we then use the variational principle to minimize the energy - the expectation value of the Hamiltonian H - with respect to the orbital coefficients cy (eq. (11.39)). This leads after a fair amount of algebra to the self-consistent Hartree-Fock equations ... 

So eq. (11.47) can be viewed as a diffusion equation in the spatial coordinates of the electrons with a diffusion coefficient D equal to j. The source and sink term S is related to the potential energy V. In regions of space where V is attractive (negative) the concentration of diffusing material (here the wavefunction) will accumulate and it will decrease where V is positive. It turns out that if we start from an initial trial wavefunction and propagate it forward in time using eq. (11.47),... 

This is referred as BO ansatz. This ansatz is taken as a variational trial function. Terms beyond the leading order in m/M are neglected m is the electronic and M is nuclear mass, respectively). The problem with expansion (4) is that functions /(r, R) contain except bound states also continuum function since it includes the centre of mass (COM) motion. Variation principle does not apply to continuum states. To avoid this problem we can separate COM motion. The remaining Hamiltonian for the relative motion of nuclei and electrons has then bound state solution. But there is a problem, because this separation mixes electronic with nuclear coordinates and also there is a question how to define molecule-fixed coordinate system. This is in detail discussed by Sutcliffe [5]. In the recent paper by Kutzelnigg [8] this problem is also discussed and it is shown how to derive adiabatic corrections using, as he called it, the Bom-Handy ansatz. There are few important steps to arrive at formula for a diabatic corrections. Firstly, one separates off COM motion. Secondly, (very important step) one does not specify the relative coordinates (which are to some extent arbitrary). In this way one arrives at relative Hamiltonian Hrd [8] with trial wavefunction If we make BO ansatz... 

A fundamental approach to computing the ground-state wavefunction and its energy for an A-electron system is the power method [20, 83]. In the power method a series of trial wavefunctions ) are generated by repeated application of the Hamiltonian... 

The Hamiltonian gradually filters the ground-state wavefunction from the trial wavefunction. To understand this filtering process, we expand the initial trial wavefunction in the exact wavefunctions of the Hamiltonian, ). With n iterations of the power method, we have... 

This upper-bound property forms the basis of the so-called variational method in which trial wavefunctions are constructed ... 

Quite often a trial wavefunction is expanded as a linear combination of other functions... 

For such a trial wavefunction, the energy depends quadratically on the linear variational Cj coefficients ... 

These permutational symmetries are not only characteristics of the exact eigenfunctions of H belonging to any atom or molecule containing more than a single electron but they are also conditions which must be placed on any acceptable model or trial wavefunction (e.g., in a variational sense) which one constructs. 

In particular, within the orbital model of electronic structure (which is developed more systematically in Section 6), one can not construct trial wavefunctions which are simple spin-orbital products (i.e., an orbital multiplied by an a or (3 spin function for each electron) such as 1 sa 1 s(32sa2s(32pia2poa. Such spin-orbital product functions must be made permutationally antisymmetric if the N-electron trial function is to be properly antisymmetric. This can be accomplished for any such product wavefunction by applying the following antisymmetrizer operator ... 

Because the total Hamiltonian of a many-electron atom or molecule forms a mutually commutative set of operators with S2, Sz, and A = (V l/N )Ep sp P, the exact eigenfunctions of H must be eigenfunctions of these operators. Being an eigenfunction of A forces the eigenstates to be odd under all Pp. Any acceptable model or trial wavefunction should be constrained to also be an eigenfunction of these symmetry operators. 

If the atom or molecule has additional symmetries (e.g., full rotation symmetry for atoms, axial rotation symmetry for linear molecules and point group symmetry for nonlinear polyatomics), the trial wavefunctions should also conform to these spatial symmetries. This Chapter addresses those operators that commute with H, Pjj, S2, and Sz and among one another for atoms, linear, and non-linear molecules. 

The most straightforward way to introduce the concept of optimal molecular orbitals is to consider a trial wavefunction of the form which was introduced earlier in Chapter 9.II. The expectation value of the Hamiltonian for a wavefunction of the multiconfigurational fomn... 

Much of the development of the previous chapter pertains to the use of a single Slater determinant trial wavefunction. As presented, it relates to what has been called the unrestricted Hartree-Fock (UHF) theory in which each spin-orbital (ftj has its own orbital energy 8i and LCAO-MO coefficients Cv,i there may be different Cv,i for a spin-orbitals than for (3 spin-orbitals. Such a wavefunction suffers from the spin contamination difficulty detailed earlier. 

Suppose the ground state solution to this problem were unknown, and that you wish to approximate it using the variational theorem. Choose as your trial wavefunction,... 

The electronic energy is a functional of the spin orbitals, and we want to minimize it subject to some set of constraints. This can be done using the calculus of variations applied to functionals. So lets look at a general example of functional variation applied to E, a functional of some trial wavefunction that can be linearly varied under a single constraint. 

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