Diabatic representation transformation

The (nearly) diagonal representation of the nuclear kinetic energy is accompanied by off-diagonal matrix elements of the potential energy in the diabatic representation. Transforming the diagonal potential energy matrix... 

It is important to note that the two surface calculations will be carried out in the diabatic representation. One can get the initial diabatic wave function matrix for the two surface calculations using the above adiabatic initial wave function by the following orthogonal transformation,... 

U(qJ is referred to as an adiabatic-to-diabatic transformation (ADT) matrix. Its mathematical sbucture is discussed in detail in Section in.C. If the electronic wave functions in the adiabatic and diabatic representations are chosen to be real, as is normally the case, U(q ) is orthogonal and therefore has n n — l)/2 independent elements (or degrees of freedom). This transformation mabix U(qO can be chosen so as to yield a diabatic electronic basis set with desired properties, which can then be used to derive the diabatic nuclear motion Schrodinger equation. By using Eqs. (27) and (28) and the orthonormality of the diabatic and adiabatic electronic basis sets, we can relate the adiabatic and diabatic nuclear wave functions through the same n-dimensional unitary transformation matrix U(qx) according to... 

Appendix C On the Single/Multivaluedness of the Adiahatic-to-Diahatic Transformation Matrix Appendix D The Diabatic Representation Appendix E A Numerical Study of a Three-State Model Appendix F The Treatment of a Conical Intersection Removed from the Origin of Coordinates Acknowledgments References... 

We have used the above analysis scheme for all single- and two-surface calculations. Thus, when the wave function is represented in polar coordinates, we have mapped the wave function, 4,ad(, t) to Tatime step to use in Eq. (17) and as the two surface calculations are performed in the diabatic representation, the wave function matrix is back transformed to the adiabatic representation in each time step as... 

The Hamiltonian in Eq. [26] is usually referred to as the diabatic representation, employing the diabatic basis set <1), Hamiltonian matrix is not diagonal. There is, of course, no unique diabatic basis as any pair obtained from (]), by a unitary transformation can define a new basis. A unitary transformation defines a linear combination of cj) and < >b which, for a two-state system, can be represented as a rotation of the (]), basis on the angle /... 

To overcome these numerical problems, one can convert to an approximately diabatic representation of the wave function via the unitary transformation [85,86] ... 

In order to solve the scattering problem, it proves convenient to avoid the sharp changes in the nonadiabatic couplings by transforming to the so-called p-diabatic representation in which the matrix elements of d/dR are zero. Explicitly, we carry out a unitary transformation using the matrix W(R) which satisfies ... 

As in section 3, the diabatic representations are obtained finding the orthogonal matrix T such that TT = P. Again, the diabatic representation is not unique because T is defined within an overall p-independent orthogonal transformation. In actual calculations, one has to manipulate the potential energy matrix V = Te(p)T, whose large dimensions are often the bottleneck in practice. Proper choice of T is therefore crucial. The practical implementation (see the final section) of hyperspherical harmonics as the proper diabatic set is of great perspective power. 

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