Wave function basis set

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

Bak, K.L., Jorgensen, P., Olsen, J., Helgaker, T., Klopper, W. Accuracy of atomization energies and reaction enthalpies in standard and extrapolated electronic wave function/basis set calculations. J. Chem. Phys. 2000,112(21), 9229-42. 

In addition to the electronically adiabatic representation described by (4) and (5) or, equivalently (57) and (58), other representations can be defined in which the adiabatic electronic wave function basis set used in expansions (4) or (58) is replaced by some other set of functions of the electronic coordinates rel or r. Let us in what follows assume that we have separated the motion of the center of mass G of the system and adopted the Jacobi mass-scaled vectors R and r defined after (52), and in terms of which the adiabatic electronic wave functions are i] l,ad(r q) and the corresponding nuclear wave function coefficients are Xnd (R). The symbol q(R) refers to the set of scalar nuclear position coordinates defined after (56). Let iKil d(r q) label that alternate electronic basis set, which is allowed to be parametrically dependent on q, and for which we will use the designation diabatic. We now proceed to define such a set. LetXn(R) be the nuclear wave function coefficients associated with those diabatic electronic wave functions. As a result, we may rewrite (58) as... 

Accuracy of atomization energies and reaction enthalpies in standard and extrapolated electronic wave function/basis set calculations ... 

All calculations presented here are based on density-functional theory [37] (DFT) within the LDA and LSD approximations. The Kohn-Sham orbitals [38] are expanded in a plane wave (PW) basis set, with a kinetic energy cutoff of 70 Ry. The Ceperley-Alder expression for correlation and gradient corrections of the Becke-Perdew type are used [39]. We employ ah initio pseudopotentials, generated by use of the Troullier-Martins scheme [40], The coreradii used, in au, were 1.23 for the s, p atomic orbitals of carbon, 1.12 for s, p of N, 0.5 for the s of H, and 1.9, 2.0, 1.5, 1.97,... 

Over a limited range of p in the neighborhood of p let us expand the symmetry wave function in the local surface function basis set (jJi Qi p)... 

Improved calculational schemes were originally developed by the American physicists John C. Slater and Conyers Herring, using plane waves as basis sets. Slater used a muffin-tin potential, where the atomic wave functions inside the atomic spheres were fit to the plane waves. He was also the first person to use a viable local exchange approximation, which he called the Xa model. 

At present, the electronic structure of crystals, for the most part, has been calculated using the density-functional theory in a plane-wave (PW) basis set. The one-electron Bloch functions (crystal orbitals) calculated in the PW basis set are delocalized over the crystal and do not allow one to calculate the local characteristics of the electronic structure. As a consequence, the functions of the minimal valence basis set for atoms in the crystal should be constructed from the aforementioned Bloch functions. There exist several approaches to this problem. The most consistent approach was considered above and is associated with the variational method for constructing the Wannier-type atomic orbitals (WTAO) localized at atoms with the use of the calculated Bloch functions. Another two approaches use the so-called projection technique to connect the calculated in PW basis Bloch states with the atomic-like orbitals of the minimal basis set. 

Density functional techniques that use plane waves as basis sets for the valence electrons and describe the atomic cores by pseudopotentials or other special techniques only became applicable to zeolites recently. This approach had already been standard in solid state physics for many years, but only recent advances in computer technology and pseudopotential theory made it possible to apply it to solids with unit cells as large as those of zeolites. Moreover, the computational efficiency of plane wave methods is coupled to the DPT approximation and the accuracy needed in chemistry is reached only with gradient-corrected functionals which emerged only in recent years. The striking advantage of plane wave basis sets is the very efficient analytical calculation of the forces on the nuclei (only Hellmann-Feynman terms contribute) which allows structure optimizations and makes even molecular dynamics (MD) runs possible by the Car-Parrinello method. This type of technique is expected to make a major contribution to zeolite chemistry over the next few years. 

It is important to note that the reliability of the results depends on a careful analysis of the many parameters to be set up and tested. The XC functional, basis sets, k points, plane waves cutoff and others details have to be set up accordingly. The chemical model of the solid/surface has to be adequately chosen to reproduce the phenomena under investigation. The reactivity of the sulfide mineral surfaces is a challenging system of technological and environmental importance, which has been recently investigated by DFT methods and will be reported in the present chapter. 

Krishnan R, Binkley J S, Seeger R and Popie J A 1980 Self-consistent molecular orbital methods XX. A basis set for correlated wave functions J. Chem. Phys. 72 650-4... 

The initial amplitudes d/(to) are obtained by projecting the initial wave function on the DVR basis set. For the initial wave function, we use... 

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

The total orbital wave function for this system is given by an electronically adiabatic n-state Bom-Huang expansion [2,3] in terms of this electronic basis set as... 

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... 

When the wave function is completely general and pennitted to vary in the entire Hilbert space the TDVP yields the time-dependent Schrodinger equation. However, when the possible wave function variations are in some way constrained, such as is the case for a wave function restricted to a particular functional form and represented in a finite basis, then the corresponding action generates a set of equations that approximate the time-dependent Schrodinger equation. 

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