Wavefunction Variational Principle

The Schrodinger equation (1.17) is equivalent to a wavefunction variational principle [2] Extremize I H I ) subject to the constraint E ) = 1, i.e., set the following first variation to zero  

The ground state energy and wavefunction are found by minimizing the expression in curly brackets. 

The unrestricted solution of (1.30) is equivalent by the method of Lagrange multipliers to the unconstrained solution of 

Since 6 is an arbitrary variation, we recover the Schrodinger equation (1.17). Every eigenstate of H is an extremum of H )/ ) and vice versa. 

The wavefunction variational principle implies the Hellmann-Feynman and virial theorems below and also implies the Hohenberg-Kohn [25] density functional variational principle to be presented later. 

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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. 

The idea of coupling variational and perturbational methods is nowadays gaining wider and wider acceptance in the quantum chemistry community. The background philosophy is to realize the best blend of a well-defined theoretical plateau provided by the application of the variational principle coupled to the computational efficiency of the perturbation techniques. [29-34]. In that sense, the aim of these approaches is to improve a limited Configuration Interaction (Cl) wavefunction by a perturbation treatment. 

HF (HF) theory is based on the idea that one takes an antisymmetrized product wavefunction and uses the variational principle to obtain the best possible approximation to the A -particle wavefunction that cannot be represented by such a single determinant. Thus, one inserts the single determinant into the Rayleigh-Ritz functional and performs a constraint variation of the orbitals. The results of the variational process are the famous HF equations that are satisfied by each of the orbitals ... 

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. 

According to the variational principle, the ground state of the system is described by those electronic wavefunctions which minimize the Kohn-Sham functional. The presence of an external perturbation is represented by a perturbation functional, Ep, that is added to the unperturbed Kohn-Sham functional ... 

The best wavefunction is that which gives the lowest value of E - this is the famous variational principle. 

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 ... 

An ansatz is then made for the time-dependent wavefunction l o (t)) depending on time-dependent coefficients which are expanded in the orders of the perturbations Ef or The wavefunction coefficients in each order of the perturbation have to be determined from the time-dependent Schrddinger equation or an equivalent time-dependent variation principle. Expressions... 

An application of the variational principle to an unbounded from below Dirac-Coulomb eigenvalue problem, requires imposing upon the trial function certain conditions. Among these the most important are the symmetry properties, the asymptotic behaviour and the relations between the large and the small components of the wavefunction related to the so called kinetic balance [1,2,3]. In practical calculations an exact fulfilment of these conditions may be difficult or even impossible. Therefore a number of minimax principles [4-7] have been formulated in order to allow for some less restricted choice of the trial functions. There exist in the literature many either purely intuitive or derived from computational experience, rules which are commonly used as a guidance in generating basis sets for variational relativistic calculations. 

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... 

The interelectronic interactions W are defined using constrained search [21, 22] over all A-representable 2-RDMs that reduce to R g). Since the set of 2-RDMs in the definition of W contains the AGP 2-RDM of g, that set is not empty and W is well defined. Through this construction, E still follows the variational principle and coincides with the energy of a wavefunction ip, which reproduces R g) = D[ T ] and = W[g]. The latter is due to... 

It is clear from the variational principle for the wavefunction that if Fq is A-representable, then Tr[HQ jvFQ] >Eg s Hf ) for every reduced Hamiltonian. To show that the converse is true, we need three key facts ... 

Relativistic variational principles are usually formulated as prescriptions for reaching a saddle point on the energy hypersurface in the space of variational parameters. The results of the variational calculations depend upon the orientation of the saddle in the space of the nonlinear parameters. The structure of the energy hypersurface may be very complicated and reaching the correct saddle point may be difficult [14,15]. If each component of the wavefunction is associated with an independent set of nonlinear parameters, then changing the representation of the Dirac equation results in a transformation of the energy hypersurface. As a consequence, the numerical stability of the variational procedure depends on the chosen representation. 

In one sense, research in theoretical chemistry at Queen s University at Kingston originated outside the Department of Chemistry when A. John Coleman came in 1960 as head of the Department of Mathematics. Coleman took up Charles Coulson s challenge150 to make the use of reduced density matrices (RDM) a viable approach to the N-electron problem. RDMs had been introduced earlier by Husimi (1940), Lowdin (1955), and McWeeny (1955). The great attraction was that their use could reduce the 4N space-spin coordinates of the wavefunctions in the variational principle to only 16 such coordinates. But for the RDMs to be of value, one must first solve the celebrated N-repre-sentability problem formulated by Coleman, namely, that the RDMs employed must be derivable from an N-electron wavefunction.151 This constraint has since been a topic of much research at Queen s University, in the Departments of Chemistry and Mathematics as well as elsewhere. A number of workshops and conferences about RDMs have been held, including one in honor of John Coleman in 1985.152 Two chemists, Hans Kummer [Ph.D. Swiss Federal Technical... 

Continuum wavefunctions can also be generated by solving the partial differential equation (3.3) directly without first transforming it into a set of ordinary differential equations. One possible scheme is the finite elements method (Askar and Rabitz 1984 Jaquet 1987). Another method, which has been applied for the calculation of multi-dimensional scattering wavefunctions, is the 5-matrix version of the Kohn variational principle (Zhang and Miller 1990). 

Approximate solutions of the time-dependent Schrodinger equation can be obtained by using Frenkel variational principle within the PCM theoretical framework [17]. The restriction to a one-determinant wavefunction with orbital expansion over a finite atomic basis set leads to the following time-dependent Hartree-Fock or Kohn-Sham equation ... 

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