Variational principle for excited states

The ADFT/ASCF-DFT scheme has been met with considerable reservation. Thus, ADFT/ASCF-DFT assumes implicitly that a transition can be represented by an excitation involving only two orbitals, an assumption that seems not generally to be satisfied. Also, the variational optimization in ASCF-DFT of the orbitals makes it difficult to ensure orthogonality between different excited state determinants when many transitions are considered, resulting ultimately in a variational collapse. Finally, it has been questioned [110] whether there exists a variational principle for excited states in DFT. In spite of this, some of the first pioneering chemical applications of DFT involved ASCF-DFT calculations on excitation energies [36, 113-116] for transition metal complexes and ASCF-DFT is still widely used [117-121]. 

In scheme (6) the basis set is optimized by invoking the variation principle for each state considered. For the ground state the optimized values of the even-tempered parameters ao and / o given by Schmidt and Ruedenberg [9] are used. We add the subscript 0 to distinguish ground state values. For the excited state optimal ai and / i values for a sequence of Mi values are determined. 

When the limiting conditions of the friction approximation are not valid, e.g., there is strong non-adiabatic coupling or rapid temporal variation of the coupling, there is at present no well-defined first principles method to calculate the breakdown in the BOA. The fundamental problem is that DFT cannot calculate excited states of adsorbates and quantum chemistry techniques, that can in principle calculate excited states, are not possible for extended systems. 

We may add that the well known difficulties of using the variation method with excited states (tn) are circumvented if xT i.e. if the first-order w.f. for excited state is a sufficient approximation to the exact w.f. of that state. Then one still has a minimum principle involving only the known, unperturbed w.f. s, of states lower than (w). 

In recent years, these methods have been greatly expanded and have reached a degree of reliability where they now offer some of the most accurate tools for studying excited and ionized states. In particular, the use of time-dependent variational principles have allowed the much more rigorous development of equations for energy differences and nonlinear response properties [81]. In addition, the extension of the EOM theory to include coupled-cluster reference fiuictioiis [ ] now allows one to compute excitation and ionization energies using some of the most accurate ab initio tools. 

It is also a common experience that traditional Cl calculations converge very poorly, because the virtual orbitals produced from an HF (or HF-LCAO) calculation are not determined by the variation principle and turn out to be very poor for representations of excited states. 

Let us now consider the possibilities for deriving an eigenfunction for a particular excited state. The straightforward application of the variation principle (Eq. II.7) is complicated by the additional requirement that the wave function Wk for the state k must be orthogonal to the exact eigenfunctions W0, Wv for all the lower states although these are not usually known. One must therefore try to proceed by way of the secular equation (Eq. III.21). A well-known theorem15 25 says that, if a truncated... 

This part introduces variational principles relevant to the quantum mechanics of bound stationary states. Chapter 4 covers well-known variational theory that underlies modern computational methodology for electronic states of atoms and molecules. Extension to condensed matter is deferred until Part III, since continuum theory is part of the formal basis of the multiple scattering theory that has been developed for applications in this subfield. Chapter 5 develops the variational theory that underlies independent-electron models, now widely used to transcend the practical limitations of direct variational methods for large systems. This is extended in Chapter 6 to time-dependent variational theory in the context of independent-electron models, including linear-response theory and its relationship to excitation energies. 

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