Born-Oppenheimer basis

We will discuss an electronic basis, the electronic basis on which the Jahn-Teller theory is based, and compare it with another electronic basis, the electronic basis. Since non-adiabatic coupling in Jahn-Teller effect has a different meaning from that in the Born-Oppenheimer basis, we will also discuss adiabatic approximations in these electronic bases. 

Consider a physically acceptable system of the complex-level structure that involves the Born-Oppenheimer basis, consisting of the ground electronic state g) = 0) and a single doorway state s) that is coupled via nonadiabatic intramolecular interactions Hvib to a background of J) states, and where both the states s) and the manifold I) are characterized by the radiative and nonradiative decay widths Ys and Yj. respectively (see Figure 6.2). The effective Hamiltonian responsible for this system is... 

B3.1.1.1 THE UNDERLYING THEORETICAL BASIS—THE BORN-OPPENHEIMER MODEL... 

The applicability of the Born-Oppenheimer approximation for complex molecular systems is basic to all classical simulation methods. It enables the formulation of an effective potential field for nuclei on the basis of the SchrdJdinger equation. In practice this is not simple, since the number of electrons is usually large and the extent of configuration space is too vast to allow accurate initio determination of the effective fields. One has to resort to simplifications and semi-empirical or empirical adjustments of potential fields, thus introducing interdependence of parameters that tend to obscure the pure significance of each term. This applies in... 

In the vibrational treatment we assumed, as usually done, that the Born-Oppenheimer separation is possible and that the electronic energy as a function of the internuclear variables can be taken as a potential in the equation of the internal motions of the nuclei. The vibrational anharmonic functions are obtained by means of a variational treatment in the basis of the harmonic solutions for the vibration considered (for more details about the theory see Pauzat et al [20]). 

The separation of the electronic and nuclear motions depends on the large difference between the mass of an electron and the mass of a nucleus. As the nuclei are much heavier, by a factor of at least 1800, they move much more slowly. Thus, to a good approximation the movement of the elections in a polyatomic molecule can be assumed to take place in the environment of the nuclei that are fixed in a particular configuration. This argument is the physical basis of the Born-Oppenheimer approximation. 

Force constants, crude Born-Oppenheimer approximation, hydrogen molecule, minimum basis set calculation, 545-550 Forward peak scattering, electron nuclear... 

NON-BORN-OPPENHEIMER VARIATIONAL CALCULATIONS OE ATOMS AND MOLECULES WITH EXPLICITLY CORRELATED GAUSSIAN BASIS EUNCTIONS... 

The symmetry requirements and the need to very effectively describe the correlation effects have been the main motivations that have turned our attention to explicitly correlated Gaussian functions as the choice for the basis set in the atomic and molecular non-BO calculations. These functions have been used previously in Born-Oppenheimer calculations to describe the electron correlation in molecular systems using the perturbation theory approach [35 2], While in those calculations, Gaussian pair functions (geminals), each dependent only on a single interelectron distance in the exponential factor, exp( pr ), were used, in the non-BO calculations each basis function needs to depend on distances between aU pairs of particles forming the system. 

Non-Bom-Oppenheimer wave functions calculated in this way look more like their Born-Oppenheimer counterparts in the smaller basis set limits, and thus a good starting guess for these may be taken from Bom-Oppenheimer calculations in the same basis. Thus we calculate the electronic part first (this requires much fewer basis functions than does a full non-Bom-Oppenhimer calculation) and then form the total basis function by multiplying each electronic portion by a guess for the nuclear portion ... 

We will present below a short description of some Born-Oppenheimer calculations we have done on this basis, followed by examples of triatomic non-Born-Oppenheimer calculations on this basis. 

A. Born-Oppenheimer Calculations in a Basis of Explicitly Correlated Gaussians... 

At this stage we are at the very beginning of development, implementation, and application of methods for quantum-mechanical calculations of molecular systems without assuming the Born-Oppenheimer approximation. So far we have done several calculations of ground and excited states of small diatomic molecules, extending them beyond two-electron systems and some preliminary calculations on triatomic systems. In the non-BO works, we have used three different correlated Gaussian basis sets. The simplest one without r,y premultipliers (4)j = exp[—r (A t (8> Is) "]) was used in atomic calculations the basis with premultipliers in the form of powers of rj exp[—r (Aj (8> /sjr])... 

All of the calculations have been performed at the experimental equilibrium distance R = 1.128 A, in order to enable a proper comparison with the EOM-CCSD reference. In so far as there are neither largely interacting excited states nor special reasons for expecting a breakdown of the Born Oppenheimer approximation, great changes in the MAE are not expected if one takes the (SC) SDCI ground state equilibrium value for Re which is Re = 1.140 A (very close to the CCSD value, as expected Cfr. table 1). We have performed a separate calculation of the whole set of VEE with the aug-cc-pVDZ basis set at the Rg distance, in any case. The results have not been included in table II for the sake of clarity, but the total MAE values where 2.34 eV for the MR-SDCI and 0.17 eV for (SC)2mR-SDCI. 

The Born-Oppenheimer approximation may then be thought of as keeping the electronic eigenfunctions independent and not allowing them to mix under the nuclear coordinates. This may be seen by expanding the total molecular wavefunction using the adiabatic eigenfunctions as a basis... 

The well-known Born-Oppenheimer approximation (BOA) assumes all couplings Kpa between the PES are identically zero. In this case, the dynamics is described simply as nuclear motion on a single adiabatic PES and is the fundamental basis for most traditional descriptions of chemistry, e.g., transition state theory (TST). Because the nuclear system remains on a single adiabatic PES, this is also often referred to as the adiabatic approximation. 

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