Bom-Oppenheimer limit

Figure 2. Effect of the frequency < > of the perturbation by the core on an electron moving in a Bohr-Sommerfeld orbit of high eccentricity (low angular momentum). Plotted vs. the angle u, which varies by 2ir over one orbit. Note that the perturbation is localized near the core. In the inverse Bom-Oppenheimer limit (x 1) the perturbation oscillates many times during one orbit of the electron. (For further details and the formalism that describes the motion at high x as diffusive-like (see Refs. 3c and S.) For higher angular momentum / the effective adiabaticity parameter is x(l - e) xfl/2, where e is the eccentricity of the Bohr-Sommerfeld orbit. States of high / are thus effectively decoupled from the core.

R. D. Levine To answer the question of Prof. Lorquet, let me say that the peaks in the ZEKE spectra correspond to the different energy states of the ion. From the beginning one was able to resolve vibrational states, and nowadays individual rotational states of polyatomics have also been resolved. The ZEKE spectrum is obtained by a (weak) electrical-field-induced ionization of a high Rydberg electron moving about the ion. The very structure of the spectrum appears to me to point to the appropriate zero-order description of the states before ionization as definite rovibrational states of the ionic core, each of which has its own Rydberg series. Such a zero-order description is inverse to the one we use at far lower energies where each electronic state has its own set of distinct rovibrational states, known as the Bom-Oppenheimer limit. 

To go beyond the Hartree-Fock limit and obtain the full solution to the Schrodinger equation (in the non-relativistic and Bom-Oppenheimer limit), one would have to combine various solutions of the product type. In any calculation one obtains more molecular orbitals than needed to accommodate all the electrons in the system. In a system with 2n electrons, the n molecular orbitals with the lowest molecular orbital energies are used in the Hartree-Fock solution for the ground state (this assumes a closed shell system, where two electrons are paired up in each molecular orbital). The rest of the molecular orbitals obtained will be excited molecular orbitals. Of course, other possible wavefunctions of the product type can be formed by using excited molecular orbitals in the product. The set of all such possible products can be used as a basis set to solve the full Schrodinger equation. The solution now looks like ... 

We refer to this equation as to the time-dependent Bom-Oppenheimer (BO) model of adiabatic motion. Notice that Assumption (A) does not exclude energy level crossings along the limit solution q o- Using a density matrix formulation of QCMD and the technique of weak convergence one can prove the following theorem about the connection between the QCMD and the BO model ... 

Having stated the limitations (non-relativistic Hamilton operator and the Bom-Oppenheimer approximation), we are ready to consider the electronic Schrodinger equation. It can only be solved exactly for the Hj molecule, and similar one-electron systems. In the general case we have to rely on approximate (numerical) methods. By neglecting relativistic effects, we also have to introduce electron spin as an ad hoc quantum effect. Each electron has a spin quantum number of 1 /2. In the presence of an... 

Corrections involving nuclei (with the nuclear spin I replacing the electron spin s) are analogous to the above one- and two-particle terms in eqs. (8.29-8.30), with the exception of those involving the nuclear mass, which disappears in the Bom-Oppenheimer approximation (which may be be considered as the Mnucieus oo limit). 

In the adiabatic limit, the coupling F is strong, so that one may consider the transition between the two quantum states a continuous motion of the system on a single Bom-Oppenheimer surface (called the adiabatic state) that is the lowest eigenvalue of the 2 x 2 matrix in Eq. (18). 

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 ... 

Having discussed ways to reduce the scope of the MCSCF problem, it is appropriate to consider the other limiting case. What if we carry out a CASSCF calculation for all electrons including all orbitals in the complete active space Such a calculation is called full configuration interaction or full CF. Witliin the choice of basis set, it is the best possible calculation that can be done, because it considers the contribution of every possible CSF. Thus, a full CI with an infinite basis set is an exact solution of the (non-relativistic, Bom-Oppenheimer, time-independent) Schrodinger equation. 

In one quantum mechanical approach based on the diabatic approximation , the electron is assumed to be confined initially at one of the reactant sites and electron transfer is treated as a transition between the vibrational levels of the reactants to those of the products. The quantum mechanical treatment begins with the time dependent Schrodinger equation, Hip = -ihSiplSt, where the wavefunction tj/ is written as a sum of the initial (reactant) and final (product) states. In the limit that the Bom-Oppenheimer approximation for the separation of electronic and nuclear motion is valid, the time dependent Schrodinger equation eventually leads to the Golden Rule result in equation (25). 

If W were known exactly, the value of a first-order property calculated from equation (12) would be exact. In practice, only an approximation to W is known, and it is important to know how the expectation value differs from the exact value. Since errors in calculated dipole moments due to the breakdown of the Bom-Oppenheimer approximation are likely to be small8 (typically 0.002 a.u.), and for most molecules relativistic effects can be ignored,6 there are two separate remaining problems in practice. The first concerns the likely accuracy when the wavefunction is at the Hartree-Fock limit, the second the effect of using a truncated basis set to obtain a wavefunction away from the Hartree-Fock limit. 

Before getting into a deeper analysis of the concept of resonance, we must define precisely what we understand by chemical structure . One of the most basic concepts in molecular quantum mechanics is the one of potential energy surface (PES). It allows us to define a molecular structure as an arrangement of nuclear positions in space. The definition of molecular structure depends on the validity of the Bom-Oppenheimer approximation for a given state. Actually, its validity is limited to selected portions of the entire Bom-Oppenheimer PES. When a state is described by one PES, we call it an adiabatic state. It is clear that the concept of chemical structure , depends on the existence of a previously defined molecular structure . Only adiabatic states have a molecular structure . From now on, we will always be dealing with adiabatic states. 

In the intramolecular situation of M+ decomposing to mj" and mJi, the vibrational frequencies and moments of inertia of the reactant are, of course, the same for each decomposition and the frequencies and moments of the transition states differ. The significant fact is that, again within the Bom — Oppenheimer approximation, the Teller — Redlich product rule applies to transition states. Choosing vibrational frequencies and moments of inertia of the transition state of one decomposition, therefore immediately fixes, within certain limits, frequencies and moments of the transition state of the other [see eqn. (29)]. The Teller — Redlich product rule has the following general form for transition states. 

Another factor of which a nonclassical theory must take account is the quantisation of the internal modes of D and A, and the consequent relaxation of the Bom-Oppenheimer constraint that the electron must transfer within a fixed nuclear framework. In classical theory, the vibrational modes of D and A are treated as classical harmonic oscillators, but in reality their quantisation is usually significant (that is, one or more of the vibration frequencies v is sufficiently high that the classical limit hv IcT does not apply). Electron transfer then requires the overlap, not only of the electronic wavefunctions of R and P, but also of their vibrational wavefunctions. It is then possible that nuclear tunnelling may assist electron transfer. As shown in Fig. 4.12, the vibrational wave-functions of R and P extend beyond the classical parabolas and overlap to some extent. This permits nuclear tunnelling from the R to the P surface, particularly in the region just below the classical intersection point. Part of the reorganisation of D and A, required prior to ET in the classical picture, may then occur simultaneously withET, by the nuclei tunnelling short (typically < 0.1 A) distances from their R to their P positions. 

Before continuing, we pause for a remark on notation. Our discussion uses a basis of zero-order states that are eigenftmctions of a Hamiltonian given by a sum of the molecular Hamiltonian in the Bom-Oppenheimer approximation and the Hamiltonian of the radiation field. As noted above, for linear spectroscopy problem we can limit ourselves to the 0- and 1-photon states of the latter. We will use... 

To start with, consider systems consisting of N dynamical electrons and positrons and K fixed nuclei with Coulomb interactions between all pairs of particles. The clamped-nuclei approximation (the Bom-Oppenheimer approximation) may be legitimate because of the huge difference in mass between electrons and nuclei. Stability of matter means that the energy of such a model system is bounded from below by a negative constant times the number of particles E —C(N 4- K). Such a condition is necessary for some basic physical properties such as the existence of the thermodynamical limit. 

ANO basis set used gives a separation in good agreement with, but smaller than, the value deduced from a combination of theory and experiment. From the convergence of the result with expansion of the ANO basis set, it is estimated that the valence limit is about 9.05 + O.lkcal/mole. The remaining discrepancy with experiment is probably mostly due to core-valence correlation effects. However, as the valence correlation treatment is nearly exact, finer effects such as Bom-Oppenheimer breakdown and relativity must also be considered. While FCI calculations have shown that a very high level of correlation treatment is required for an accurate estimate of the CV contribution to the separation, theoretical calculations indicate that CV correlation will increase the separation by at most 0.35 kcal/mole (see later discussion). Therefore, it is now possible to achieve an accuracy of considerably better than one kcal/mole in the singlet-triplet separation in methylene. 

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