Eigenfunctions approximate

The majority of photochemistry of course deals with nondegenerate states, and here vibronic coupling effects aie also found. A classic example of non-Jahn-Teller vibronic coupling is found in the photoelection spectrum of butatiiene, formed by ejection of electrons from the electronic eigenfunctions [approximately the molecular orbitals). Bands due to the ground and first... 

Ploehn and Russel (1989) developed a matched asymptotic solution of Eq. (73) equivalent to a two-eigenfunction approximation for Gc. Near the surface, interchain interactions distort chain configurations so that the characteristic length is /. Far from the surface, chains are ideal and the characteristic length scales as nl/2l l. The widely separated length scales enable the inner ground state solution to be matched asymptotically to an outer solution, yielding a uniform approximation for all z. 

In the remaining chapters, when we solve the Schrodinger equation in a canonically orthogonal basis, we will find that a further rotation of the canonical vectors is required to construct the eigenfunction approximations in the chosen basis set. 

Note that h is simply the diagonal matrix of zeroth-order eigenvalues In the following, it will be assumed that the zeroth-order eigenfunction a reasonably good approximation to the exact ground-state wavefiinction (meaning that Xfi , and h and v will be written in the compact representations... 

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

In this work, relativistic effects are included in the no-pah or large component only approximation [13]. The total electronic Hamiltonian is H (r R) = H (r R) + H (r R), where H (r R) is the nom-elativistic Coulomb Hamiltonian and R) is a spin-orbit Hamiltonian. The relativistic (nomelativistic) eigenstates, are eigenfunctions of R)(H (r R)). Lower (upper)... 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

To obtain the Hamiltonian at zeroth-order of approximation, it is necessary not only to exclude the kinetic energy of the nuclei, but also to assume that the nuclear internal coordinates are frozen at R = Ro, where Ro is a certain reference nucleai configuration, for example, the absolute minimum or the conical intersection. Thus, as an initial basis, the states t / (r,s) = t / (r,s Ro) are the eigenfunctions of the Hamiltonian s, R ). Accordingly, instead of Eq. (3), one has... 

One of the things illustrated by this calculation is that a surprisingly good approximation to the eigenvalue can often be obtained from a combination of approximate functions that does not represent the exact eigenfunction very closely. Eigenvalues are not vei y sensitive to the eigenfunctions. This is one reason why the LCAO approximation and Huckel theory in particular work as well as they do. 

According to (3.95), CLTST approximates Cf(t) by the -function, being, in a sense, a zerotime limit of the first of the equations (3.101). In the representation of eigenfunctions n>, n > the flux-flux correlation function equals... 

A test of the accuracy of the approximate wave function 0 for the ground state has been given by Eckart3 by considering the mean square deviation from the exact eigenfunction lP0 ... 

Excited states of the hydrogen molecule may be formed from a normal hydrogen atom and a hydrogen atom in various excited states.2 For these the interelectronic interaction will be small, and the Burrau eigenfunction will represent the molecule in part with considerable accuracy. The properties of the molecule, in particular the equilibrium distance, should then approximate those of the molecule-ion for the molecule will be essentially a molecule-ion with an added electron in an outer orbit. This is observed in general the equilibrium distances for all known excited states but one (the second state in table 1) deviate by less than 10 per cent from that for the molecule-ion. It is hence probable that states 3,4, 5, and 6 are formed from a normal and an excited atom with n = 2, and that higher states are similarly formed. 

The potential function obtained is only approximately correct, for the eigenfunctions are in fact largely perturbed by the inter-electronic interaction. There are no forces tending to molecule... 

In certain cases one of these might approximate the correct eigenfunction... 

The value o+l <0.4 found for H2 shows that even in the lowest state the molecules are rotating freely, the intermolecular forces producing only small perturbations from uniform rotation. Indeed, the estimated (3vq<135° corresponds to Fo <28 k, which is small compared with the energy difference 164 k of the rotational states j = 0 and j= 1, giving the frequency with which the molecule in either state reverses its orientation. The perturbation treatment shows that with this value of Fo the eigenfunctions and energy levels in all states closely approximate those for the free spatial rotator.9... 

In ordinary crystalline hydrogen there are three molecules with j = 1 for every one with j = 0. The eigenfunctions for these molecules approximate those for free molecules, namely... 

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