Hamiltonian pseudopotential

Let us suppose an infinite nondegenerate polymer chain (e.g., polythiophene) doped heavily with electron acceptors. At a high dopant content, the polymer-chain structure and electronic structure of the doped polymer are radically different from those of the intact polymer. As typical cases, we will describe two kinds of lattice structures of doped polythiophene (dopant content, 25 mole% per thiophene ring) a polaron lattice and a bipolaron lattice. They are the regular infinite arrays of polarons and bipolarons. The schematic polymer-chain structures are shown in Figure 4-16. Band-structure calculations have been performed for polaron and/or bipolaron lattices of poly(p-phenylene) [124], polypyrrole [124], polyaniline [125], polythiophene [124, 126], and poly( p-phenylenevinylene) [127], with the valence-effective Hamiltonian pseudopotential method on the basis of geometries obtained by MO methods. The schematic electronic band structures shown in Figure 4-17... 

The systems discussed in this chapter give some examples using different theoretical models for the interpretation of, primarily, UPS valence band data, both for pristine and doped systems as well as for the initial stages of interface formation between metals and conjugated systems. Among the various methods used in the examples are the following semiempirical Hartree-Fock methods such as the Modified Neglect of Diatomic Overlap (MNDO) [31, 32) and Austin Model 1 (AMI) [33] the non-empirical Valence Effective Hamiltonian (VEH) pseudopotential method [3, 34J and ab initio Hartree-Fock techniques. 

All calculations are scalar relativistic calculations using the Douglas-Kroll Hamiltonian except for the CC calculations for the neutral atoms Ag and Au, where QCISD(T) within the pseudopotential approach was used [99], CCSD(T) results for Ag and Au are from Sadlej and co-workers, and Cu and Cu from our own work, using an uncontracted (21sl9plld6f4g) basis set for Cu [6,102] and a full active orbital space. 

Hamiltonian which, in a nonlocal pseudopotential approximation, can be written as... 

In this paper, the effect of the pseudopotential term, arising from the quantum mechanical correction to classical mechanism (V ), on the torsional levels of hydrogen peroxide and deuterium peroxide is evaluated. The V operator, depends on the first and second derivatives with respect to the torsional coordinate of the determinant of the g inertia matrix and on the first derivatives of the B kinetic energy parameter of the vihrational Hamiltonian. V has heen determined for each nuclear conformation from the optimized coordinates obtained using MP2/AUG-cc-pVTZ ah initio calculations. 

As is well known, the vibrational Hamiltonian defined in internal coordinates may be written as the sum of three different terms the kinetic energy operator, the Potential Energy Surface and the V pseudopotential [1-3]. V is a kinetic energy term that arises when the classic vibrational Hamiltonian in non-Cartesian coordinates is transformed into the quantum-mechanical operator using the Podolsky trick [4]. The determination of V is a long process which requires the calculation of the molecular geometry and the derivatives of various structural parameters. 

SRPA has been already applied for atomic nuclei and clusters, both spherical and deformed. To study dynamics of valence electrons in atomic clusters, the Konh-Sham functional [14,15]was exploited [7,8,16,17], in some cases together with pseudopotential and pseudo-Hamiltonian schemes [16]. Excellent agreement with the experimental data [18] for the dipole plasmon was obtained. Quite recently SRPA was used to demonstrate a non-trivial interplay between Landau fragmentation, deformation splitting and shape isomers in forming a profile of the dipole plasmon in deformed clusters [17]. 

The generalization of the pseudopotential method to molecules was done by Boni-facic and Huzinaga[3] and by Goddard, Melius and Kahn[4] some ten years after Phillips and Kleinman s original proposal. In the molecular pseudopotential or Effective Core Potential (ECP) method all core-valence interactions are approximated with l dependent projection operators, and a totally symmetric screening type potential. The new operators, which are parametrized such that the ECP operator should reproduce atomic all electron results, are added to the Hamiltonian and the one electron ECP equations axe obtained variationally in the same way as the usual Hartree Fock equations. Since the total energy is calculated with respect to this approximative Hamiltonian the separability problem becomes obsolete. 

The scheme for achieving the core-valence separation which we wish to discuss in this Report is embodied in the idea of a pseudopotential . The hamiltonian for the electronic part of the wavefunction can be symbolically expressed within the Bom-Oppenheimer ( clamped nucleus ) approximation as... 

The first and last terms in equation (34) consist of the valence-electron components of the all-electron hamiltonian of equation (1), and the remaining terms constitute the pseudopotential represented symbolically in equation (2). Further, if we assume that the interaction of the two cores A and B can be approximated by a point charge potential (see Kahn et al.2i for errors in this assumption),... 

We can see that the non-uniqueness of the pseudopotential and of the open-shell hamiltonian have similar origins. Following Roothaan36 the total open-shell hamiltonian may be written in terms of the basic operator Pa by using projection operators to define the particular form of the operator for each sub-space ... 

However, this is definitely the technique for future calculations involving a large number of metal atoms. Furthermore, the idea behind the pseudopotential method is also applied in other types of Hamiltonians described below, e.g., valence effective Hamiltonian and semi-empirical methods. 

Hess, B. A. Dolg, M. Relativistic Quantum Chemistry with Pseudopotentials and Transformed Hamiltonians, Relativistic Effects in Heavy-Element Chemistry and Physics -, Ed. Hess, B. A. Wiley Chichester, 2002, pp. 89-122. 

Further uses of pseudopotentials are numerous. The most obvious (and rather widely known ones) are to continue with the PP or MP Hamiltonians for a widely understood combination of the core and valence shells and to apply standard ab initio techniques to electrons in the valence subspace only. We do no elaborate further on this as the hybrid nature of the pseudopotential methods is rather obvious from the above and its more specific applications in a narrower QM/MM hybrid context will be described later. 

An efficient way to solve a many-electron problem is to apply relativistic effective core potentials (RECP). According to this approximation, frozen inner shells are omitted and replaced in the Hamiltonian hnt by an additional term, a pseudopotential (UREP)... 

Pseudo-potential energy curves are extracted from the perturbative Hamiltonian of Eq. (17) by retaining only the terms with P = N = 0—that is, the terms without differential operator. One obtains a one-dimensional pseudopotential curve Fvi.vsCQ) for each pair of quantum numbers v (H-CN stretch) and V3 (C-N stretch)... 

The remarkable conclusion of this argument is that though pseudopotentials can be used to describe semiconductors as well as metals, the pseudopotential perturbation theory which is the essence of the theory of metals is completely inappropriate in semiconductors. Pseudopotenlial perturbation theory is an expansion in which the ratio W/Ey of the pseudopotential to the kinetic energy is treated as small, whereas for covalent solids just the reverse quantity, Ey/W, should be treated as small. The distinction becomes /wimportant if we diagonalize the Hamiltonian matrix to obtain the bands since, for that, we do not need to know which terms are large. Thus the distinction was not essential to the first use of pseudopotentials in solids by Phillips and KIcinman (1959) nor in the more recent application of the Empirical Pseudopotenlial Method u.scd by M. L. Cohen and co-workers. Only in approximate theories, which are the principal subject of this text, must one put terms in the proper order. 

Extension of pseudopotential theory to the transition metals preceded the use of the Orbital Correction Method discussed in Appendix E, but transition-metal pseudopotentials are a special case of it. In this method, the stales are expanded as a linear combination of plane waves (or OPW s) plus a linear combination of atomic d states. If the potential in the metal were the same as in the atom, the atomic d states would be eigenstates in the metal and there would be no matrix elements of the Hamiltonian with other slates. However, the potential ix different by an amount we might write F(r), and there arc, correspondingly, matrix elements (k 1 // 1 r/> = hybridizing the d states with the frce-eleclron states. The full analysis (Harrison, 1969) shows that the correct perturbation differs from (5K by a constant. The hybridization potential is... 

It will be useful later to notice that the core states arc eigenstates of the Hamiltonian, so that // 11> can be replaced by c>, but we can most clearly see the appropriateness of the empty-core model of the pseudopotential by leaving /f as a kinetic and potential energy. Wc have seen that the pscudopotential always enters our calculations in a matrix element between plane waves, so wc write such a matrix clement as... 

The pseudopotential concept was advanced a long ago [1] and is based on the natural energetic and spatial separation of core and valence electrons. The concept allows a significant reduction in computational efforts without missing the essential physics of phenomena provided the interaction of core and valence electrons is well described by some effective (model) Hamiltonian. Traditionally, pseudopotentials are widely used in the band structure calculations [2], because they allow convenient expansions of the wavefunctions in terms of plane waves suited to describing periodical systems. For molecular and/or nonperiodical systems, the main advantage of pseudopotentials is a... 

Within the density functional theory (DFT), several schemes for generation of pseudopotentials were developed. Some of them construct pseudopotentials for pseudoorbitals derived from atomic calculations [29] - [31], while the others make use [32] - [36] of parameterized analytical pseudopotentials. In a specific implementation of the numerical integration for solving the DFT one-electron equations, named Discrete-Variational Method (DVM) [37]- [41], one does not need to fit pseudoorbitals or pseudopotentials by any analytical functions, because the matrix elements of an effective Hamiltonian can be computed directly with either analytical or numerical basis set (or a mixed one). 

These pseudopotentials are inconvenient because the resulting pseudoorbitals are eigenfunctions to different Hamiltonians. We developed a more suitable scheme [44], [45] based on the Christiansen-Lee-Pitzer approach [9]. Let us consider first the construction of pseudopotentials, then their use for molecular systems, and, finally, for simulating surfaces and bulk of solids. 

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