Time-dependent variation theory

In practice, we cannot solve the time-dependent Schrodinger equation 

To allow for the presence of a rapidly oscillating phase factor, the normalized wavefunction (which we now denote by V,) may be written as 

When H is time-independent, (12.2.5a) reduces to the usual stationary-state variation principle, while (12.2.4a) yields the corresponding phase factor where E is the stationary value of the usual energy 

On adopting the tensor notation of Section 2.4, these last two expressions yield 

On substituting (12.2.10) and (12.2.8) in (12.2.5), and remembering that f o is ih time-independent variation function corresponding to we obtain 

---

This part introduces variational principles relevant to the quantum mechanics of bound stationary states. Chapter 4 covers well-known variational theory that underlies modern computational methodology for electronic states of atoms and molecules. Extension to condensed matter is deferred until Part III, since continuum theory is part of the formal basis of the multiple scattering theory that has been developed for applications in this subfield. Chapter 5 develops the variational theory that underlies independent-electron models, now widely used to transcend the practical limitations of direct variational methods for large systems. This is extended in Chapter 6 to time-dependent variational theory in the context of independent-electron models, including linear-response theory and its relationship to excitation energies. 

In recent years, these methods have been greatly expanded and have reached a degree of reliability where they now offer some of the most accurate tools for studying excited and ionized states. In particular, the use of time-dependent variational principles have allowed the much more rigorous development of equations for energy differences and nonlinear response properties [81]. In addition, the extension of the EOM theory to include coupled-cluster reference fiuictioiis [ ] now allows one to compute excitation and ionization energies using some of the most accurate ab initio tools. 

Obviously, the BO or the adiabatic states only serve as a basis, albeit a useful basis if they are determined accurately, for such evolving states, and one may ask whether another, less costly, basis could be Just as useful. The electron nuclear dynamics (END) theory [1-4] treats the simultaneous dynamics of electrons and nuclei and may be characterized as a time-dependent, fully nonadiabatic approach to direct dynamics. The END equations that approximate the time-dependent Schrddinger equation are derived by employing the time-dependent variational principle (TDVP). 

Time-Dependent Variational Principle in Density Functional Theory... 

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. 

Wigner rotation/adiabatic-to-diabatic transformation matrices, 92 Electronic structure theory, electron nuclear dynamics (END) structure and properties, 326-327 theoretical background, 324-325 time-dependent variational principle (TDVP), general nuclear dynamics, 334-337 Electronic wave function, permutational symmetry, 680-682 Electron nuclear dynamics (END) degenerate states chemistry, xii-xiii direct molecular dynamics, structure and properties, 327 molecular systems, 337-351 final-state analysis, 342-349 intramolecular electron transfer,... 

Weiner, B. and Trickey, S.B. (1999). Time-dependent variational principle in density functional theory, Adv. Quantum Chem. 35, 217-247. 

Burton BP, Kikuchi R (1984) The antiferromagnetic-paramagnetic transition in a-Fc203 in the single prism approximation of the cluster variation method. Phys Chem Minerals 11 125-131 Carpenter MA, Salje E (1989) Time-dependent Landau theory for order/disorder processes in minerals. Mineral Mag 53 483-504... 

The END theory was proposed in 1988 [11] as a general approach to deal with time-dependent non-adiabatic processes in quantum chemistry. We have applied the END method to the study of time-dependent processes in energy loss [12-16]. The END method takes advantage of a coherent state representation of the molecular wave function. A quantum mechanical Lagrangian formulation is employed to approximate the Schrodinger equation, via the time-dependent variational principle, by a set of coupled first-order differential equations in time to describe the END. 

Let us assume (a) that the energy levels of our molecules are E0, Elt E%, . En (b) that the fraction of molecules in the with state at time t is xm(t) (c) that the transition probabilities per unit time Wnm from state m to n can be computed in terms of the interaction of the molecules with a heat bath (which is postulated to remain at temperature T) by application of quantum-mechanical time dependent perturbation theory (the fVBra s being proportional to squares of absolute values of the matrix elements of the interaction energy) and (d) that the temporal variations of the level concentrations are described through the transport equation... 

If the time variation of this equation is taken at the first stages, the coefficients ak(t) can be nearly equal to the initials. For any k value different to n, we can consider that at t = 0, the ak 0) = 0 and an 0)= 1, whereas at f / 0, ak(t) 0 and u (t) 1. In this time-dependent perturbation theory, the time at which the perturbation occurs is small. Since the energy e/iov < 0.1 eV, because of Heisenberg s principle, t< h/eEoy)K 1013 s. Then, it is admissible to consider a (t) 1. On the other hand, we take into account that only one state mostly contributes to the sum and with this arbitrary approximation, we restrict the series to only one component ... 

The form of Jf, given by Eq. (3), is not very helpful with regard to understanding the relationship of Jf to H. There are many different ways in which to formulate this problem, in terms of time-independent perturbation theory [8,9,13-15], time-dependent perturbation theory [8,9,16,17], the coupled-cluster or e method [18], moment methods [19], and variational approaches [20]. Because of time and space restrictions, we will discuss in detail only the time-independent perturbation-theory approach. Those interested in other techniques should peruse the appropriate references. 

---