Non-adiabatic corrections

The leading quantum correction to the static JT energetics is given by the zero-point energy gain due to the softening of the vibrational frequency at the JT-distorted minima [5]. To obtain this information, by finite differences we compute the Hessian matrix of the second-order derivatives of the lowest adiabatic potential sheet, at one of the static JT minima Q,mn 

The first two columns fix the relevant distortion. The third column indicates the spin S of the excited states considered. The excitation energies in the last two columns are referred to the adiabatic energy of each specific minimum, reported in Table 3 

The following column contains the leading nonadiabatic correction zer0, the zero-point energy defined in equation (14). The last column reports the adiabatic energy Eclass corrected by the zero-point term zero for n 4 it leads to a different ordering of the spin states 

In the present calculation both e-e and e-v interactions are included for the HOMO shell of C6o- E-e exchange terms are treated essentially exactly, in the assumptions that (i) inter-band couplings can be neglected, and only act as a renormalization of the Coulomb parameters and that (ii) the latter are independent of the charge n in the HOMO. In principle, due to both orbital and geometrical relaxation, the effective 

The present calculation was carried out in the linear e-v approximation. As the coupling and thus the distortions are fairly large, quadratic and higher-order (in Q) couplings and vibrations anharmonicity could be important. Unfortunately, no estimate for those higher-order couplings is available yet. 

---

The simplest way to add a non-adiabatic correction to the classical BO dynamics method outlined above in Section n.B is to use what is known as surface hopping. First introduced on an intuitive basis by Bjerre and Nikitin [200] and Tully and Preston [201], a number of variations have been developed [202-205], and are reviewed in [28,206]. Reference [204] also includes technical details of practical algorithms. These methods all use standard classical trajectories that use the hopping procedure to sample the different states, and so add non-adiabatic effects. A different scheme was introduced by Miller and George [207] which, although based on the same ideas, uses complex coordinates and momenta. 

The non-adiabatic corrections to the electronic energy are represented by the term A HF,... 

Non-adiabatic correction to zero - particle term of the fermionic Hamiltonian. Correction to the fermionic ground state energy... 

Figure 11 2 6. The non-adiabatic correction to the fermionic ground state energy/umt cell of the MgB2 as the function of the parameter q - see text. For q =2, it is —49.4 meV...

Figure 11 2 7. The orbital energies of the unoccupied states after the non-adiabatic corrections (shifts) v.s. the original-uncorrected orbital energies of the o2 band (left) and 71 band (right). The minimum on the graphs indicates the energy position of the lowest unoccupied state of the 02 band (left) and n band (right) with respect to EF (=0)...

From the Figs.7. one can see that after the non-adiabatic corrections, the energy distance of the lowest unoccupied state from the EF (half-gap) is 7.6 meV for a2 band and 4 meV for n band. 

Figure 11 2 8. The final - non-adiabatic density of the unoccupied states at 0 K, according to Eq.19.The uncorrected density of states is normalized to 1 and EF = 0. The peak at 4 meV corresponds to non-adiabatic corrections of n band states and the peak at 7.6 meV corresponds to non-adiabatic corrections of a2 band states. The density of the occupied states is the mirror picture with respect to EF...

Generally speaking, the adiabatic wave function (2) is not a stationary one because it is not the eigen function of total Hamiltonian of the system (1). In reality, the electron wave function J/M(r R) depends on R and so the differential operator rR acts not only on / (R), but also on i/q/r R). It results in appearance of non-adiabatic correction operator in the basis of functions (2)... 

The Bom-Oppenheimer approximation is not always correct, especially with light nuclei and/or at finite temperature. Under these circumstances, the electronic distribution might be less well described by the solution of a Schroedinger equation. Non-adiabatic effects can be significant in dynamics and chemical reactions. Usually, however, non-adiabatic corrections are small for equilibrium systems at ordinary temperature. As a consequence, it is generally assumed that nuclear dynamics can be treated classically, with motions driven by Bom-Oppenheimer potential energy functions ... 

The Bom-Oppenheimer Principle in the adiabatic approximation is one among many approximations that are made in molecular theory. The question of how precise is the approximation is not often asked and is difficult to answer. It is often brushed aside. One way to proceed is to compare exact solutions of the full molecular Hamiltonian with solutions found in the adiabatic approximation. It should be possible to test whether the latter solutions approach the former as non-adiabatic corrections are added. It is necessary to work with a Hamiltonian and associated Schrodinger equation that is capable of exact solution. 

The problem has not been resolved analytically. Thirunamachandran and I showed that in special cases answers can be given. If we suppose that both electronic and vibrational motions are represented as simple harmonic vibrations, and the coupling between them given a sufficiently simple form, then the full Hamiltonian can be solved exactly to find energies and eigenfunctions. These exact solutions can be compared with those found in the adiabatic approximation with non-adiabatic corrections. 

It is well known that (4.3) can be solved exactly, but to make the comparison it is better to solve (4.3) in a perturbation expansion, and take the adiabatic result with non-adiabatic correction. We find agreement at least to terms quadratic in the coupling constant c [186]. It is further possible to show that the adiabatic approximation goes to an asymptotic limit that is not precisely equal to the exact solution, but very close to it even in pathological conditions. 

Model Calculations testing the Adiabatic Bom-Oppenheimer Approximation and its non-Adiabatic Corrections. 

J. Chem. Phys., 103, 7277-7286 (1995) d) S. D. Schwartz, The Interaction Representation and Non-Adiabatic Corrections to Adiabatic Evolution Operators,/. Chem. Phys., 104, 1394-1398 (1996) e) D. Antoniou, S. D. Schwartz, Nonadiabatic Effects in a Method that Combines Classical and Quantum Mechanics, /. Chem. Phys., 104, 3526-3530 (1996) f) S. D. Schwartz, The Interaction Representation and Non-Adiabatic Corrections to Adiabatic Evolution Operators II Nonlinear Quantum Systems, /. 

S. D. Schwartz, The Interaction Representation and Non-Adiabatic Corrections to Adiabatic Evolution Operators, /. Chem. Phys., 104, 1394— 1398 (1996). 

The modem theory of chemical reaction is based on the concept of the potential energy surface, which assumes that the Born-Oppenheimer adiabatic approximation [16] is obeyed. However, in systems subjected to the Jahn-Teller effect, adiabatic potentials have the physical meaning of the potential energy of nuclei only under the condition that non-adiabatic corrections are small [28]. In the vicinity of the locally symmetric intermediate, these corrections will be very large. The complete description of nuclear motion, i.e. of the mechanism of the chemical reaction, can be obtained only from Schroedinger s equation without applying the Born-Oppenheimer approximation in the vicinity of the locally... 

This chapter deals exclusively with the ground state problem. Even this task has to be simplified further for a treatment of most of the realistic cases one has to rely on the adiabatic approximation in which the electronic subsystem is treated in the frame of nuclear positions at rest. For many aspects, relativistic corrections are more important than non-adiabatic corrections. Generally, the former inerease with increasing nuclear charge while the latter decrease with increasing nuclear mass. The adiabatic electronic states (groxond state and quasi-particles) constitute what is commonly called the electronic structure of a solid. 

As to many-electron systems, corrections to the BO approximation can be obtained by means of a second-order contact transformation method [28]. This introduces two terms (a) the simple DBOC, which gives rise to a mass-dependent correction to the PES and (b) the considerably more difficult second-order (also called non-adiabatic) correction, which introduces coupling between electronic states and primarily results in corrections to the kinetic energy operator. In the most sophisticated first-principles treatments [17,34,35] allowance is made for non-adiabatic effects though further work is required to explore the best possible strategies for compufafion and utilization of this information. 

The approach reported is able to produce the non-adiabatic corrections to all rovibrational levels corresponding to the ground electronic state. ... 

Le Roy et al. [OSRloy] give an alternative treatment of the frequency data (Direct-potential fit) which is based on the eigenvalues of the radial Schrddinger equation. The potential is expressed in form of a modified Morse function where the exponent is written in form of a power series expansion. The coefficients and those in the non-adiabatic correction terms are determined in a direct fit to the experimental eigenvalue differences, see [05Roy] for details and results. 

---