Extended Born-Oppenheimer equation for tri-state system

The Journal of Physics and Chemistry of Solids, 2008

The ARPES of high-T c cuprates and theoretical results of low-Fermi energy band structure fluctuation for different groups of superconductors indicate that electron coupling to pertinent phonon modes drive system from adiabatic into anti-adiabatic state (o4E F). At these circumstances, not only Migdal-Eliashberg approximation is not valid, but basic adiabatic Born-Oppenheimer approximation (BOA) does not hold. At these circumstances, electronic structure has to be studied as explicitly dependent on instantaneous nuclear coordinates Q as well as on instantaneous nuclear momenta P. In the present paper-part I, it has been shown that Q, P-dependent modification of the BOA for ground electronic state can be derived by sequence of canonical transformations of the basis functions. The effect of nuclear coordinates and momenta on electronic structure is presented in the form of corrections to zero-, one-and two-particle terms of clamped nuclear Hamiltonian. In the anti-adiabatic state, correction to electronic ground state energy (zero-particle term correction) is negative and system can be stabilized in the anti-adiabatic state at distorted geometry with respect to adiabatic equilibrium structure and gap in one-particle spectrum of quasi-continuum states at Fermi level can be opened. Stabilization effect is solely the consequence of nuclear dynamics (P) that is crucial in anti-adiabatic state. It has been shown that nuclear dynamics also increases electron correlation until system at nuclear motion remains in a bound state. Corresponding corrections to electronic wave function are also specified. On the other hand, when system remains at vibration motion of nuclei in adiabatic state, the influence of nuclear dynamics (P-dependence) is negligible. In this case, all basic effects are covered through nuclear coordinates (Q-dependence) within the adiabatic BOA and standard results of solidstate (or molecular) physics are recovered.

The Journal of Chemical Physics, 2009

When a set of three states is coupled with each other but shows negligibly weak interaction with other states of the Hilbert space, these states form a sub-Hilbert space. In case of such subspace [J. Chem. Phys. 124, 074101 (2006)], (a) the adiabatic-diabatic transformation (ADT) condition, ∇⃗A+τ⃗A=0 [Chem. Phys. Lett. 35, 112 (1975)], provides the explicit forms of the nonadiabatic coupling (NAC) elements in terms of electronic basis function angles, namely, the ADT angles, and (b) those NAC terms satisfy the so-called curl conditions [Chem. Phys. Lett. 35, 112 (1975)], which ensure the removal of the NAC elements [could be singular also at specific point(s) or along a seam in the configuration space] during the ADT to bring the diabatic representation of the nuclear Schrödinger equation with a smooth functional form of coupling elements among the electronic states. Since the diabatic to adiabatic representation of the Hamiltonian is related through the same unitary transformation ...

Physical Review A, 2000

In this work is applied the extended-approximated Born-Oppenheimer ͑BO͒ equation, as derived in the preceding article ͓Phys. Rev. A 62, 032506, ͑2000͔͒ to a tristate system. For this sake an appropriate scattering two-arrangement-two-coordinate model was devised. The calculations for each energy were done three times: once without doing any approximation, and two times by approximating the three coupled Schrödinger equations by two different, but gauge-related, single ͑extended͒ BO equation. State-to-state ͑reactive and nonreac-tive͒ transition probabilities were obtained, indicating that the new extended approximate BO equations yield relevant results for a tristate system.

Communications in Mathematical Physics, 2001

We reconsider the time-dependent Born-Oppenheimer theory with the goal to carefully separate between the adiabatic decoupling of a given group of energy bands from their orthogonal subspace and the semiclassics within the energy bands. Band crossings are allowed and our results are local in the sense that they hold up to the first time when a band crossing is encountered. The adiabatic decoupling leads to an effective Schrödinger equation for the nuclei, including contributions from the Berry connection.

The Journal of Physical Chemistry A, 2003

In this article, we discuss the electronic nonadiabatic coupling matrix, τ, which under certain conditions is characterized by two interesting features: (1) its components fulfill an extended Curl equation (Chem. Phys. Lett. 1975, 35, 112, (see Appendix 1)) and it is quantized in the sense that the topological D matrix, presented as an exponentiated line integral over the τ matrix, is a unitary diagonal matrix (Chem. Phys. Lett. 2000, 319, 489). These features can be shown to exist if the relevant group of states forms a Hilbert subspace, namely, a group of states that are strongly coupled with each other but are only weakly coupled with all other states. The numerical study is carried out applying the eigenfunctions of the Mathieu equation. † Part of the special issue "Donald J. Kouri Festschrift".

2021

A system of two bilinearly coupled harmonic oscillators has been solved analytically by using the Born-Oppenheimer (BO) product wavefunction ansatz and the phasespace bound trajectory approach [J. S. Molano et al., Chem. Phys. Lett. 76(12), 138171 (2021)]. The bilinearly coupled oscillator system allows to obtain the analytical expression of the quantum system, facilitating comparison with the results of using the BO ansatz product. The analytical and BO wavefunctions are obtained as a product of parabolic cylinder functions. The arguments of the parabolic cylinder functions of the exact and the BO wavefunctions are related to the physical coordinates by linear transformations, represented by matrices G and G̃ respectively. A QU decomposition of the matrix G outputs an upper triangular matrix U that is closely related to the BO G̃ matrix. The effect of the BO non-adiabatic coupling is analyzed on the stability of the phase-space equations of motion. The eigenvalues and eigenfunction...

Journal of Chemical Theory and Computation, 2020

The major bottleneck of first principle based beyond Born−Oppenheimer (BBO) treatment originates from large number and complicated expressions of adiabatic to diabatic transformation (ADT) equations for higher dimensional sub-Hilbert spaces. In order to overcome such shortcoming, we develop a generalized algorithm, "ADT" to generate the nonadiabatic equations through symbolic manipulation and to construct highly accurate diabatic surfaces for molecular processes involving excited electronic states. It is noteworthy to mention that the nonadiabatic coupling terms (NACTs) often become singular (removable) at degenerate point(s) or along a seam in the nuclear configuration space (CS) and thereby, a unitary transformation is required to convert the kinetically coupled (adiabatic) Hamiltonian to a potentially (diabatic) one to avoid such singularity(ies). The "ADT" program can be efficiently used to (a) formulate analytic functional forms of differential equations for ADT angles and diabatic potential energy matrix and (b) solve the set of coupled differential equations numerically to evaluate ADT angles, residue due to singularity(ies), ADT matrices, and finally, diabatic potential energy surfaces (PESs). For the numerical case, user can directly provide ab initio data (adiabatic PESs and NACTs) as input files to this software or can generate those input files through in-built python codes interfacing MOLPRO followed by ADT calculation. In order to establish the workability of our program package, we selectively choose six realistic molecular species, namely, NO 2 radical, H 3 + , F + H 2 , NO 3 radical, C 6 H 6 + radical cation, and 1,3,5-C 6 H 3 F 3 + radical cation, where two, three, five and six electronic states exhibit profound nonadiabatic interactions and are employed to compute diabatic PESs by using ab initio calculated adiabatic PESs and NACTs. The "ADT" package released under the GNU General Public License v3.

Physical Review A, 1993

A general method to decouple multicomponent linear wave equations is presented. First, the Weyl calculus is used to transform operator relations into relations between c-number valued matrices. Then it is shown that the symbol representing the wave operator can be diagonalized systematically up to arbitrary order in an appropriate expansion parameter. After transforming the syn~bols back to operators, the original problem is reduced to solving a set of linear uncoupled scalar wave equations. The procedure is exemplified for a particle with a Born-Oppenheimer-type Hamiltonian valid through second order in fi. The resulting effective scalar Harniltonians are seen to contain an additional velocitydependent potential. This contribution has not been reported in recent studies investigating the adiabatic motion of a neutral particle moving in an inhomogeneous magnetic field. Finally, the relation of the general method to standard quantum-mechanical perturbation theory is discussed. PACS numberk): 03.65.Sq, 03.40.Kf

Chemical Physics Letters, 1983

Following an original idea of Bacr, since then several times applied to various small molecular collisional systems. it is shown that-whatever the choice of Curvilinear generalized coordinates to describe the molecular gometry and owxall rotation-the algebraic fmmework proposed by Baer remains valid provided only that the esnct tensorhl expression of the kinetic energy operator is used.

ESAIM: Mathematical Modelling and Numerical Analysis, 2007

We explain why the conventional argument for deriving the time-dependent Born-Oppenheimer approximation is incomplete and review recent mathematical results, which clarify the situation and at the same time provide a systematic scheme for higher order corrections. We also present a new elementary derivation of the correct second-order time-dependent Born-Oppenheimer approximation and discuss as applications the dynamics near a conical intersection of potential surfaces and reactive scattering.