Partial waves energy derivatives

By using the above derived expressions in the partial-wave form we can write the self energy as... 

The partial-wave methods do, however, have two distinct advantages. Firstly, they provide solutions of arbitrary accuracy for a muffin-tin potential and, for close-packed systems, this makes them far more accurate than any traditional fixed-basis method. Secondly, the information about the potential enters (1.21) only via a few functions of energy, the logarithmic derivatives aln i/ (E,r) /aln r, at the muffin-tin sphere. 

The linear methods devised by Andersen [1.19] are characterised by using fixed basis functions constructed from partial waves and their first energy derivatives obtained within the muffin-tin approximation to the potential. 

These methods therefore lead to secular equations (1.21) which are linear in energy, that is to eigenvalue equations of the form (1.19). When applied to a muffin-tin potential they use logarithmic-derivative parameters and provide solutions of arbitrary accuracy in a certain energy range. The linear methods thus combine the desirable features of the fixed-basis and partial-wave methods. 

The linear muffin-tin orbital (LMTO) method described in detail in the following chapters employs a fixed basis set in the form of muffin-tin orbitals (MTO). A muffin-tin orbital is everywhere continuous and differentiable and, inside the MT spheres it is constructed from the partial waves ijj (E, r), and their first energy derivatives... 

Both methods include a procedure for correcting some of the approximations made in the interstitial region. The ASW employs an elegant technique in which the energy derivatives of the structure constants must be calculated. Similarly, the LMTO method requires an extra set of structure constants, but these will correct both for the approximations in the interstitial region and for the neglect of higher partial waves. 

In Chap.5 we derive the LCMTO equations in a form not restricted to the atomic-sphere approximation, and use the , technique introduced in Chap.3 to turn these equations into the linear muffin-tin orbital method. Here we also give a description of the partial waves and the muffin-tin orbitals for a single muffin-tin sphere, define the energy-independent muffin-tin orbitals and present the LMTO secular matrix in the form used in the actual programming, Sect.9.3. 

By analogy we make the following definitions of the partial wave and its energy derivative, cf. (3.4,5),... 

Although the errors introduced by the atomic-sphere approximation are unimportant for many applications, e.g. self-consistency procedures, there are cases where energy bands of high accuracy are needed, and where one should include the perturbation (6.2) in some form. Below, we derive an expression which accounts to first order for the differences between the sphere, atomic or muffin tin, and the atomic polyhedron, re-establishes the correct kinetic energy in the region between the sphere and the polyhedron, and corrects for the neglect of higher partial waves. The extra terms added to the LMTO matrices which accomplish these corrections are called the combined correction terms [6.2]. 

The fourth step is to construct a new trial charge density. To this end the programme evaluates the moments of the projected state densities and inserts these together with the partial waves and their energy derivatives into (6.41, 8.30). The moment calculation is based on a set of projected state densities obtained in previous runs of LMTO and DDNS where the potential parameters used in LMTO could be those given in step three, i.e. constructed from a renormalised atomic potential, in a previous separate run of SCFC. The moments are evaluated by... 

The asymptotic behavior of the second-order energy of the M0ller-Plesset perturbation theory, especially adapted to take advantage of the closed-shell atomic structure (MP2/CA), is studied. Special attention is paid to problems related to the derivation of formulae for the asymptotic expansion coefficients (AECs) for two-particle partial-wave expansions in powers... 

Furthermore, the energy derivative of D, in terms of the partial-wave amplitude at the ASA sphere boundary is known. 

The partial structure factors for binary (Bhatia and Thorton, 1970) and multicomponent (Bhatia and Ratti, 1977) liquids have been expressed in terms of fluctuation correlation factors, which at zero wave number are related to the thermodynamic properties. An associated solution model in the limits of nearly complete association or nearly complete dissociation has been used to illustrate the composition dependence of the composition-fluctuation factor at zero wave number, Scc(0). For a binary liquid this is inversely proportional to the second derivative of the Gibbs energy of mixing with respect to atom fraction. 

Where h is Planck s constant over 2n h = 1.05 x 10 J), D, is the partial derivative, here with respect to time, iFis the wave function of the system, and H is the Hamiltonian, an energy operator which in the molecular case may be written... 

The nth-order elastic coefficients may be defined as the nth partial derivatives of the energy. The higher-order coefficients (n > 3) should be included when the propagation of a finite amplitude waves (i.e., nonlinear phenomena) is under consideration. 

When the external electric field is time-dependent, there is no well-defined energy of the molecular system in accordance with Eq. (100), and the wave function response can thus not be retrieved from a variational condition on the energy as in the analytic derivative approach described above. Instead the response parameters have to be determined from the time-dependent Schrodinger equation, a procedure which was illustrated in Section 3 for the exact state case. In approximate state theories, however, our wave function space only partially spans the 7V-electron Hilbert space, and the response functions that correspond to an approximate state wave function will clearly be separate from those of the exact state wave function. This fact is disregarded in the sum-over-states approach, and, apart from the computational aspect of slowly converging SOS expressions, it is of little concern when highly accurate wave function models are used. But for less flexible wave function models, the correct response functions should be used in the calculation of nonlinear optical properties. 

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