Variational method states

The complexity of molecular systems precludes exact solution for the properties of their orbitals, including their energy levels, except in the very simplest cases. We can, however, approximate the energies of molecular orbitals by the variational method that finds their least upper bounds in the ground state as Eq. (6-16)... 

The Seetion entitled The BasiC ToolS Of Quantum Mechanics treats the fundamental postulates of quantum meehanies and several applieations to exaetly soluble model problems. These problems inelude the eonventional partiele-in-a-box (in one and more dimensions), rigid-rotor, harmonie oseillator, and one-eleetron hydrogenie atomie orbitals. The eoneept of the Bom-Oppenheimer separation of eleetronie and vibration-rotation motions is introdueed here. Moreover, the vibrational and rotational energies, states, and wavefunetions of diatomie, linear polyatomie and non-linear polyatomie moleeules are diseussed here at an introduetory level. This seetion also introduees the variational method and perturbation theory as tools that are used to deal with problems that ean not be solved exaetly. 

Most of the techniques described in this Chapter are of the ab initio type. This means that they attempt to compute electronic state energies and other physical properties, as functions of the positions of the nuclei, from first principles without the use or knowledge of experimental input. Although perturbation theory or the variational method may be used to generate the working equations of a particular method, and although finite atomic orbital basis sets are nearly always utilized, these approximations do not involve fitting to known experimental data. They represent approximations that can be systematically improved as the level of treatment is enhanced. 

For the accurate, a priori calculation of reaction rates, variational transition state calculations are now the method of choice. These calculations are capable of giving the highest-accuracy results, but can be technically dilficult to perform... 

UFF (universal force field) a molecular mechanics force field unrestricted (spin unrestricted) calculation in which particles of different spins are described by different spatial functions VTST (variational transition state theory) method for predicting rate constants... 

Even if the process were at steady state, tensile strength, a key property would still reflect some variation. Steady state, or stable operation of any process, has associated with it a characteristic variation. Superimposed on this is the testing method, which is itself a process with its own characteristic variation. The observed variation is a composite of these two variations. 

Schwartz, H. M., Phys. Rev. 103, 110, "Ground state solution of the non-relativistic equation for He." More rapid convergence in the Ritz variational method by inclusion of half-integral powers in the Hylleraas function. 

Berencz, F., Acta Phys. Hung. 6, 423, Calculation of the ground state of H2 on the basis of the variation method. ... 

The variation method gives an approximation to the ground-state energy Eq (the lowest eigenvalue of the Hamiltonian operator H) for a system whose time-independent Schrodinger equation is... 

As a simple application of the variation method to determine the ground-state energy, we consider a particle in a one-dimensional box. The Schrodinger equation for this system and its exact solution are presented in Section 2.5. The ground-state eigenfunction is shown in Figure 2.2 and is observed to have no nodes and to vanish at x = 0 and x = a. As a trial function 0 we select 0 = x(a — x), 0 X a... 

If we use the wave function for the unperturbed ground state as a trial function 0 in the variation method of Section 9.1 and set H equal to then we have from equations (9.2), (9.18), and (9.24)... 

In this section we examine the ground-state energy of the helium atom by means of both perturbation theory and the variation method. We may then compare the accuracy of the two procedures. 

As a normalized trial function 0 for the determination of the ground-state energy by the variation method, we select the unperturbed eigenfunction r2) of the perturbation treatment, except that we replace the atomic number Zby a parameter Z ... 

The variation method is usually employed to determine an approximate value of the lowest eneigy state (the ground state) of a given atomic or molecular system. It can, furthermore, be extended to the calculation of energy levels of excited stales. It forms the basis of molecular orbital theory and that which is often referred to (incorrectly) as theoretical chemistry". 

Allison TC, Trahlar DG (1998) Testing the accuracy of practical semiclassical methods variational transition state theory with optimized multidimensional tunnelling. In Thompson DL (ed) Modern Methods for Multidimensional Dynamics Computations in Chemistry. World Scientific, Singapore, p 618... 

Table 6.2 Tests of Variational Transition State Theory by Comparing with Exact Quantum Calculations (Extracted from Allison, T. C. and Truhlar, D. G. Testing the accuracy of practical semiclassical methods variational transition state theory with optimized multidimensional tunneling, in Thompson, D. L., Ed. Modem methods for multidimensional dynamics computations in chemistry, World Scientific, Singapore 1998. pp 618-712. This reference quotes results on many more reactions and BO surfaces over broad temperature ranges.)The numbers in the table are ratios of the results of the approximate calculation to the quantum calculation, all at 300 K...

Being the lowest stable excited state, the electronic structure of the B state of H2 has been of considerable interest. The calculation of Kolos and Wolniewicz using the variational method with elliptic coordinates [57] showed that the wavefunction is well represented by a mixture of three configurations ionic,... 

The most satisfactory situation for making an extrapolation of rate data to the true threshold arises when the threshold is uncertain, but we can confidently calculate the functional form of the rate-energy curve from accurate kinetic theory. For small systems, it is feasible to calculate dissociation rates by quantum methods, but this is not yet feasible for the systems of interest to us. Various approaches to variational transition-state theory (VTST) provide classical or semiclassical calculations that are feasible for large systems and seem to be accurate when carefully... 

The ACSE has important connections to other approaches to electronic structure including (i) variational methods that calculate the 2-RDM directly [36-39] and (ii) wavefunction methods that employ a two-body unitary transformation including canonical diagonalization [22, 29, 30], the effective valence Hamiltonian method [31, 32], and unitary coupled cluster [33-35]. A 2-RDM that is representable by an ensemble of V-particle states is said to be ensemble V-representable, while a 2-RDM that is representable by a single V-particle state is said to be pure V-representable. The variational method, within the accuracy of the V-representabihty conditions, constrains the 2-RDM to be ensemble N-representable while the ACSE, within the accuracy of 3-RDM reconstruction, constrains the 2-RDM to be pure V-representable. The ACSE and variational methods, therefore, may be viewed as complementary methods that provide approximate solutions to, respectively, the pure and ensemble V-representabihty problems. 

Variational methods based on the use of explicitly correlated wavefiinctions are reviewed. Different types of such functions are considered. Application of explicitly correlated functions as basis functions in variational calculations on two-, three- and four-electrons molecules is presented. The state of art calculations and future perspectives are briefly discussed. 

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