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Atomic overlap matrix

Elements of the atomic overlap matrix (AOM) for atom k are given by... [Pg.188]

CF are the matrices of the molecular orbital coefficients, S is the atomic overlap matrix and e are diagonal matrices of the KS orbital eigenvalues. Now the difference between HF and KS approaches resides in the term which is... [Pg.49]

Eq. (4) provides a simple and direct formalism to obtain the Fukui function from an approach based on a relationship with the FMO s. The condensed Fukui function for electrophilic (nucleophilic) attack involves the HOMO (LUMO) FMO coefficients (c) and the atomic overlap matrix elements (S). [Pg.333]

For first- and seeond-row atoms, the Is or (2s, 2p) or (3s,3p, 3d) valenee-state ionization energies (aj s), the number of valenee eleetrons ( Elee.) as well as the orbital exponents (es, ep and ej) of Slater-type orbitals used to ealeulate the overlap matrix elements Sp y eorresponding are given below. [Pg.198]

The integrals involving atomic orbitals are often collected together into a matrix called the overlap matrix S... [Pg.103]

Think of ethene, where we use tt basis functions and Xa- if identify these as ordinary atomic 2p orbitals, then we can calculate the overlap matrix using the methods described earlier. I will write it as... [Pg.143]

The variational problem is to minimize the energy of a single Slater determinant by choosing suitable values for the MO coefficients, under the constraint that the MOs remain orthonormal. With cj) being an MO written as a linear combination of the basis functions (atomic orbitals) /, this leads to a set of secular equations, F being the Fock matrix, S the overlap matrix and C containing the MO coefficients (Section 3.5). [Pg.314]

Minimizing the total energy E with respect to the MO coefficients (see Refs. 2 and 3) leads to the matrix equation FC = SCE (where S is the overlap matrix). Solving this matrix is called the self-consistent field (SCF) treatment. This is considered here only on a very approximate level as a guide for qualitative treatments (leaving the more quantitative considerations to the VB method). The SCF-MO derivation in the zero-differential overlap approximations, where overlap between orbitals on different atoms is neglected, leads to the secular equation... [Pg.28]

The formal vector cp (K) denotes the set of atomic orbital basis functions with centers at the original nuclear locations of the macromolecular nuclear configuration K, where the components cp(r, K) of vector q(K) are the individual AO basis functions. The macromolecular overlap matrix corresponding to this set cp (K) of AO s is denoted by S(K). The new macromolecular basis set obtained by moving the appropriate local basis functions to be centered at the new nuclear locations is denoted by cfcK ), where the notation cp(r, K ) is used for the individual components of this new basis set overlap matrix is denoted by S(K ). [Pg.74]

The calculation of the indices requires the overlap matrix S of atomic orbitals and the first-order density (or population) matrix P (in open-shell systems in addition the spin density matrix Ps). The summations refer to all atomic orbitals /jl centered on atom A, etc. These matrices are all computed during the Hartree-Fock iteration that determines the molecular orbitals. As a result, the three indices can be obtained... [Pg.306]

Since we are essentially interested in qualitative features, we choose the canonical ir orbitals to be the Hiickel-Wheland orbitals.50) These are 7r-molecular orbitals which are expressed as linear combinations of the (2pz) atomic orbitals on the various carbon atoms. If the latter are denoted by px, p2, p3. .. then their overlap matrix is assumed to be S1)... [Pg.57]

Near computational linear dependence in the basis set was monitored in all calculations reported in this paper by diagonalizing the overlap matrix. A 30s basis subset was centred on each of the points defining a particular distributed basis set. Diffuse basis fimctions were deleted from off-atom basis sets until the smallest eigenvalue of the overlap matrix, e, satisfied the condition e < 10 . So, for example, the basis set designated 30s ac 28s oa ac) [nj = 5] which arises in... [Pg.163]

We emphasize that here we are speaking of orthogonalizing the VB basis not the underlying atomic orbitals (AOs). This can be accomplished by a transformation of the overlap matrix to convert it to the identity... [Pg.19]

Pt surfaces tend to restructure into overlayers with an even higher density of Pt atoms than the close-packed (111) surface [21]. The Pt atoms are closer to each other on the reconstructed surfaces than in the (111) surface. The overlap matrix elements and hence the bandwidth are therefore larger, the d bands are lower and consequently these reconstructed surfaces bind CO even weaker than the (111) surface. The reconstructed Pt surfaces are examples of strained overlayers. The effect of strain can be studied theoretically by simply straining a slab. Examples of continuous changes in the d band center and in the stability of adsorbed CO due to strain are included in Figure 4.10. The effect due to variations in the number of layers of a thin film of one metal on another can also be described in the d band model [22,23]. [Pg.271]

The matrix element is understood to be on-the-energy-shelF, i.e., the energy e of the photoelectron has to be calculated according to equ. (1.29a). Due to the different binding energies of electrons ejected from different shells of the atom, it is therefore possible to restrict the calculation of the matrix element to the selected process in the present example to photoionization in the Is shell only. As a consequence, the matrix element factorizes into two contributions, a matrix element for the two electrons in the Is shell where one electron takes part in the photon interaction, and an overlap matrix element for the other electrons which do not take part in the photon interaction (passive electrons). The overlap matrix element is given by... [Pg.47]

In the frozen atomic structure approximation, where the same orbitals are used in the initial and final states, this overlap matrix element yields unity. Hence, one obtains for the remaining matrix element... [Pg.47]


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See also in sourсe #XX -- [ Pg.188 ]




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Overlap matrix

Overlapping atoms

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