Approximations adiabatic

The adiabatic approximation is a simplification widely used in the theory of elementary processes. 

it seems expedient to formulate the adiabatic approximation in a general form without definite specification of the degrees of freedom which this approximation concerns. 

The adiabatic approximation consists in the separation of coordinates of subsystems with substantially different inherent velocities. 

Suppose that all degrees of freedom of the system can be divided into two totalities characterized by the coordinates q and Q and belonging to two subsystems with strongly different characters of motion. The characteristic velocities of the subsystem with coordinates q (referred to below as the fast subsystem) are assumed to be much larger than those of the subsystem with coordinates Q (called below slow subsystem). Accordingly, the total Hamiltonian H will be represented in the form 

The slow subsystem coordinates in this equation represent the parameters on which both the adiabatic functions 9. and the eigenvalues referred to as adiabatic terms of the fast subsystem), depend. The next step of the adiabatic approximation is the assumption that the motion of the slow subsystem does not change the fast subsystem wave function 9, i.e. that the fast subsystem follows the slow one without inertia. Mathematically, this is expressed taking the total wave function of the system Tj as the product of the adiabatic wave function 9[jt.(q Q) of the fast subsystem and the wave function of X[iz(Q ) of the slow subsystem  

The first-order approximation to eq. (6), the so-called adiabatic approximation, consists of neglecting the off-diagonal matrix elements of C (i.e. Cpp t=0 when F r ). It can be shown 7) that, for the groimd state of H2+, the adiabatic correction accoimts for 99.8% of the total correction to the B.O. approximation. A similar statement is correct (8,9) for the ground state of H2. We expect that the adiabatic correction is sufficient for all tiie systems considered here and subsequent discussion will be restricted to the adiabatic correction. 

In order to simplify the notation, the subscripts on E (R), F(R), % and C are hencefortii omitted. Then, in the adiabatic approximation. 

It should be noted that C is in general a function of R since is a function of R. F is the rotational-vibrational wavefunction of the molecule and eq. (9) is the Schroedinger equation for this wavefunction. The potential energy term in eq. (9) is the sum of E (R), the B.O. electronic energy as a function of R, which is independent of nuclear masses, and the adiabatic correction C, which does depend on nuclear masses. It is clear that the subsequent interest of the present discussion centers on C, which is sometimes referred to as the diagonal nuclear motion 

ACS Symposium Series American Chemical Society Washington, DC, 1975. 

F(R) may be approximated in the usual manner (10) Iqr the producTof a vibrational and a rotational wavefimction 

We could expect that the Born-Oppenheimer separation of electronic and nuclear motions will provide a not quite satisfactory approximation if the nuclei can move far away from their equilibrium positions which is really the case in their excited vibrational states  

This problem has been investigated by DAUDEL and BRATOZ /24/, using the perturbation theory, in order to take into account the electron-nuclear interactions which are neglected in the adiabatic approximation. For this purpose, expression (9.1) can be written in the form 

Expression (10.1) represents a system of coupled equations for all possible electronic states m. Neglecting all coupling terms, i.e., setting = 0 (and c = 0), leads again to the usual Born- 

Oppenheimer approximation given by (6.1). An exact solution of (10.1), which should give the coefficients j (x) in (8.1) and the total energy E, is very difficult however, perturbation theory may be used, provided all c are small. Therefore, we can write 

It can be shown /24/ that the first-order approximation gives / 0 but E a 0, while the second-order perturbation of the wave 

that the operator H lf, depends on the length of the vector R, but not on its direction.  

If the curves Ej(R) for different / are well separated on the energy scale, we may expect that the coupling between them is small, and therefore all 0 / for A / may be set equal to zero. This is called the adiabatic approximation. In this approximation we obtain from (6.14)  

The function fic(R) depends explicitly not only on R, but also on the direction of vector R, and therefore will describe future vibrations of the molecule (changes of i ) as well as its rotations (changes of the direction of R). 

Let us stop for a while to catch the sense of the adiabatic approximation. 

This parallel fails in one important point the Alps do not move in the potential created by tourists, the dominant geological processes are tourist-independent. As we will soon see, nuclear motion is dictated by the potential which is the electronic energy. 

Let s consider a molecule composed of N nuclei and p electrons, within a reference system, connected to the molecule s mass center (CM) for which we will collectively note  

For the atomic case, for example of a nucleus of mass (M) and an electron of mass (/w), once the energy equipartition (as in the classical mechanics) is accepted, then the (quantum) energies of the two particles become equal  

This idea is transposed to the molecular systems under the perturbation form  

This way the stationary state equation (i.e., the stationary Schrodinger s equation) for the molecule is writing in the general form  

In the zeroth order approximation (of the fixed-nuclei) the kinetic energy of the nuclei disappears, so remaining the so-called electronic equation 

Using (1.3-6) show that (1.7) is obtained, provided that the terms 

Hint Write the kinetic energy operator of the ions in the form 

The first sum in the neglected terms quoted above is responsible for the scattering of electrons by the thermal vibrations, giving rise to resistance to electronic transport [1.33]. 

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One common approximation is to separate the nuclear and electronic degrees of freedom. Since the nuclei are considerably more massive than the electrons, it can be assumed that the electrons will respond mstantaneously to the nuclear coordinates. This approximation is called the Bom-Oppenlieimer or adiabatic approximation. It allows one to treat the nuclear coordinates as classical parameters. For most condensed matter systems, this assumption is highly accurate [11, 12]. 

Pack R T and Hirschfelder J O 1970 Energy corrections to the Born-Oppenheimer approximation. The best adiabatic approximation J. Chem. Phys. 52 521-34... 

The familiar BO approximation is obtained by ignoring the operators A completely. This results in the picture of the nuclei moving over the PES provided by the electrons, which are moving so as to instantaneously follow the nuclear motion. Another common level of approximation is to exclude the off-diagonal elements of this operator matrix. This is known as the Bom-Huang, or simply the adiabatic, approximation (see [250] for further details of the possible approximations and nomenclature associated with the nuclear Schrodinger equation). 

THE CRUDE BORN-OPPENHEIMER ADIABATIC APPROXIMATION OF MOLECULAR POTENTIAL ENERGIES... 

This means that the electronic and nuclear wave functions cannot be separated anymore, and therefore the adiabatic approximation cannot be applied beyond the second-order perturbation. 

THE CRUDE BORN—OPPENHEIMER ADIABATIC APPROXIMATION 521 Consider the integral... 

THE CRUDE BORN-OPPENHEIMER ADIABATIC APPROXIMATION 555 which is a more general foiin of Eq. (131). The modification is simple ... 

Thus, the neglect of the off-diagonal matrix elements allows the change from mixed states of the nuclear subsystem to pure ones. The motion of the nuclei leads only to the deformation of the electronic distribution and not to transitions between different electronic states. In other words, a stationary distribution of electrons is obtained for each instantaneous position of the nuclei, that is, the elechons follow the motion of the nuclei adiabatically. The distribution of the nuclei is described by the wave function x (R i) in the potential V + Cn , known as the proper adiabatic approximation [41]. The off-diagonal operators C n in the matrix C, which lead to transitions between the states v / and t / are called operators of nonadiabaticity and the potential V = (R) due to the mean field of all the electrons of the system is called the adiabatic potential. 

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