Wavefunction adiabatic

For a general non-adiabatic wavefunction, the average (expectation) values of nuclear coordinates can similarly serve to define the point on the surface. ... 

To consider the nature of this approximation one should notice that the nuclear kinetic energy operator acts both on the electronic and the nuclear parts of the BO wavefunction. Hence, the deviations from the adiabatic approximation will be measured by the matrix elements of the nuclear kinetic energy, T(Q), and of the nuclear momentum. The approximate adiabatic wavefunctions have the following off-diagonal matrix elements between different vibronic states ... 

The sudden changes in the adiabatic wavefunctions near avoided crossings make it more convenient to use diabatic potential energy surfaces when simulating photodissociation dynamics. The adiabatic potentials, usually constructed from electronic structure calculation data, should therefore be transformed to diabatic potentials. The adiabatic-diabatic transformation yields diabatic states for which the derivative couplings above approximately vanish. The diabatic potential energy surfaces are obtained from the adiabatic ones by a unitary orthogonal transformation [22,23]... 

We note that, for CT transitions, the expression for /fab is not needed if the Mulliken-Hush approach is used to calculate H h from experimental quantities as discussed in Section 1.3.4. Also, the generalized Mulliken-Hush treatment [32, 33] allows the calculation of //ab from the adiabatic wavefunctions and the complete Hamiltonian the extension of Eq. 56 to include more than two states is then used to obtain Hub-... 

One could argue that this makes DFT unsuitable for the location of all MECPs and the description of spin-state crossing regions in general. This is because the adiabatic wavefunctions at the MECP are inherently multireferenfial in nature, since they are mixtures of diabatic states of different spin. However,... 

If the deperturbed curves intersect and are characterized by very different molecular constants, then they are diabatic curves (see Fig. 3.5). If the crossing is avoided, adiabatic curves are involved, and one of these curves can have a double minimum. One frequently finds that, in the region of an avoided crossing, the adiabatic wavefunction changes electronic character abruptly and the derivative of the electronic wavefunction with respect to R can be large. In... 

Fig. 8. Illustration of the changes in the content of an adiabatic wavefunction and of a qualitative diabatic reading. Case of the state of Na. ...

More generally, the integral may also equal a multiple of tt in view of the trigonometric functions in Eq. (9), and the diabatic wavefunctions (10) still be well defined. A value of tt is expected when encircling a conical intersection between potential energy surfaces in order to compensate for the singularity of the adiabatic wavefunctions (see also Chapters 1 and 7 in this book). In the ideal case, Eq. (22) is generalized as ... 

As illustrated also in other Chapters of this book (see especially Chapters 3), the transformation from the adiabatic wavefunctions computed by ah initio methods to diabatic ones, is often the first task to be accomplished when approaching the study of the dynamics around a CI. The main historical reason for the preference given to the diabatic representation was the problem of the divergence of the coupling between adiabatic surfaces, as illustrated for example in the review of Ref. 1. This computational difficulty seems to have been overcome, judging from the increasing... 

In the context of conical intersections, the concept of the geometrical or Berry phase is often discussed with the claim that an adiabatic wavefunction would change its sign circulating around a conical intersection. However, the generality of this phase change is not given it is a possible, but not a mandatory feature. 

Following the line of arguments in Ref. 50, we will therefore illustrate the conditions for the occurrence of such a phase jump, going back to the transformation between diabatic and adiabatic wavefunctions ... 

The adiabatic wavefunction is dominated by different diabatic states in the two subspaces separated by the hypersurface = 0. 

Fig. 6. Example of closed pathways around a conical intersection. In two dimensions (left), the space of degeneracy reduces to a single point and a phase jump is observed for the corresponding adiabatic wavefunction. In the three-dimensional example (right) the space of degeneracy of the two hypersurfaces = 0 (sphere) and = 0 (plane)...

Along an arbitrarily shaped closed path around a conical intersection, the adiabatic wavefunction only changes sign if the hypersurface = 0 is passed an odd number of times on each side of = 0. 

Equation (2.13) constitutes the Born-Oppenheimer approximation. When the electronic adiabatic wavefunctions are chosen real, the non-adiabatic term simply reads A = G = ( 6 T ). Usually, A is very small and Eq.(2.13) is only used if very high accuracy is sought. Neglecting A leads to the so-called adiabatic approximation... 

Nafie, L.A. (1983) Adiabatic behavior beyond the Born-Oppenheimer approximation. Complete adiabatic wavefunctions and vibrationally induced electronic current density. J. Chem. Phys., 79, 4950-4957. 

The adiabatic states, hereinafter labeled by the superscript (A), are uniquely defined through equation (10), apart from arbitrary phase factors. The Bom-Oppenheimer approximation now consists in neglecting the dynamic couplings and T kl - eigenvalues Ek take the place of the matrix elements H kk as potential energy functions. The derivative matrix elements between adiabatic wavefunctions (the nonadiabatic couplings) obey the Hellman-Feynman-like formula ... 

Several methods to determine quasi-diabatic states are based on the assumption that single determinants or configurations are quasi-diabatic wavefunctions in other words, their physical content should not change too much with the nuclear geometry and the matrix elements of d/dQa in this basis should be small. When this is true, the main source of variation of the adiabatic wavefunctions, and the dominant contribution to is the change of the Cl coefficients as functions of the nuclear coordinates keeping the Cl coefficients as constant as... 

The diagonal matrix elements ( p Al/ p ) are the effective potential energy surface that governs nuclear motion. From Equations 1.10 and 1.23, it is evident that the vibrational wavefunction x differs from the adiabatic wavefunction x As long as the basis set

electronic space, the CA basis is perfectly adequate (independent of the choice of qg). The two matrix representations 1.8 and (1.20) are merely two different representations of the same operator. 

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