L. Lathouwers

On the equivalence of time-dependent variational-principles

Abstract We consider the relationship between three variational principles which generate approximate time evolution in a parametrized manifold of wavefunctions. These are: the McLachlan variational principle, the Dirac-Frenkel variational principle and the time-dependent variational principle. We show that if the manifold can be parametrized by pairs of “complementary” parameters, the abovementioned principles are equivalent. The condition of complementarity, which is a sufficient one, is demonstrated to be satisfied in a number of important applications. However, a case of non-equivalence in the recent literature warns against liberal assumptions about the equivalence of the time-dependent variational principles.

Quantum theory and molecular spectra

Abstract The “generator coordinate method” is proposed as a fully quantum mechanical treatment of molecular spectra.

Quantum time evolution of vibrational states in curve-crossing problems

Abstract The quantum time evolution in a two-state curve-crossing situation can be computed with the split-operator FFT method. We present an improved version of that scheme, based on the analytic exponentiation of two-by-two matrices. We use this approach to investigate the vibrational dynamics of the coupled b′, c′ 1 Σ + u states of the N 2 molecule.

Dynamics of wave packets and the time-dependent variational principle

Abstract The quantal time evolution of a one-dimensional Gaussian wave packet according to the time-dependent variational principle (TDVP) is shown to be formally equivalent to the classical motion of a particle in a two-dimensional potential. The formal and practical advantages of the TDVP approach are indicated.

Applications of the generator coordinate approximation to diatomic systems. I. The hydrogen molecular ion

Calculations in the generator coordinate approximation are presented for vibration–rotation levels of the hydrogen molecular ion in its ground electronic state. The numerical results confirm general theoretical predictions made earlier and demonstrate that the generator coordinate approximation is a workable nonadiabatic approach to molecules.

The generator coordinate approximation for molecules: A review

The generator coordinate approximation is a non-adiabatic theory of molecular systems. Its fundamental outlines were developed during the 1970s. A further analysis and first applications were published during the 1980s. In this paper, we review the present status of the theory.

Perturbation analysis of the generator coordinate approximation

A Born–Oppenheimer analysis of the generator coordinate approximation is performed. The order of magnitude of generator coordinate nonadiabatic effects is estimated and their dependence on the electron‐nuclear coupling is established. A separation of the nonadiabatic effects into vibrational and rotational contributions reveals their behavior with respect to vibrational and rotational quantum numbers.

Applications of the generator coordinate approximation to diatomic systems. III, Curve-crossing problems

The generator coordinate approximation, previously applied to vibration–rotation levels near potential‐energy minima, is now worked out for curve‐crossing situations. We define the weak and strong adiabatic coupling limits. For weak adiabatic coupling both the adiabatic and generator coordinate approximations become exact. In the strong adiabatic coupling limit the adiabatic approximation breaks down, whereas the generator coordinate approximation again reproduces the exact solutions. These theoretical results are confirmed by calculations for a Hamiltonian modeled to the EF,GK 1Σ+g curve crossing in the electronic spectrum of the hydrogen molecule.

Time-dependent variational principles and conservation laws in wavepacket dynamics

A criterion for the equivalence of time-dependent variational principles in current use in chemistry and physics is demonstrated. Conservation laws are considered and it is shown that under certain conditions quantal conservation laws also hold for the time evolution derived from the variational principle. The formalism is applied to an N-dimensional spherical Gaussian wavepacket. A numerical example for the three-dimensional spherical Gaussian potential well is worked out to illustrate the dynamics and the role of conservation laws.

On the theory of non-adiabatic effects in molecules

Abstract It is shown that the generator coordinate approximation introduces non-adiabatic effects of the correct sign (energy lowering) and size (Born-Oppenheimer analysis). The theoretical expression applied to diatomic molecules qualitatively explains the observed trends of non-adiabatic energy corrections both in the adiabatic region of the spectrum and in the region of avoided crossings.