Lowdin basis

The chromophore atoms are assigned charges given by the density matrix in the Lowdin basis from the INDO/S calculation ... 

The two-electron integrals in the Lowdin basis are then assumed to be given by... 

The commutation rules for operators over the nonorthogonal set % can be derived in a straightforward manner by substituting this inverse transformation and utilizing the commutation rules in the Lowdin basis ... 

Fig. 15.3. Change in the block structure of one-electron integrals in Lowdin basis upon R cx). The labels a, b refer to the original overlapping basis set, while i, k denote Lowdin-orthogonalized orbitals. The two bases are in a one-to-one correspondence...

Since the zeroth-order equations are solved only in the Hartree-Fock approximation, the perturbation corrections account not only for intermolec-ular interactions, but for the intramolecular correlation energy as well. These two effects cannot be separated in the Lowdin basis set, but one may subtract the contributions of those [ij kl] integrals which result in local correlation. 

It may be of some interest to check whether the BSSE-free interaction operator of Eq. (15.27) could be used to solve this problem. This project was carried out in our laboratory (Surjan et al. 1985b, Surjan Poirier 1986). To utilize the full power of this interaction operator, we did not turn to a Lowdin basis, but applied a non-Hermitian perturbation theory similar to the former work by Kochanski and Gouyet (1975). 

The Lowdin population analysis scheme was created to circumvent some of the unreasonable orbital populations predicted by the Mulliken scheme, which it does. It is different in that the atomic orbitals are first transformed into an orthogonal set, and the molecular orbital coefficients are transformed to give the representation of the wave function in this new basis. This is less often used since it requires more computational work to complete the orthogonalization and has been incorporated into fewer software packages. The results are still basis-set-dependent. 

The Mulliken scheme suffers from all of the above, while the Lowdin method solves problems (1), (2) and (3). In the orthogonalized basis all off-diagonal elements are 0, and the diagonal elements are restricted to values between 0 and 2. 

A certain answer may be found (Lowdin 1955) by considering the first-order density matrix y(x x1) defined by Eq. II.9. By means of the basis y>k and formula III. 14, this matrix may be expressed in the form... 

The basis AO adopted may be Slater-type AO 11), Gaussian AO 12>, and Hartree-Fock AO 13>, Lowdin s orthogonalized AO 14>, and so on. In many cases the Slater AO s for the valence-shell electrons are taken. Clementi has extended the basis beyond the valence shells 15>. Frequently, the exponents of Slater AO s are optimized. Clementi has also adopted two different variable exponents for "one Slater AO 15>. 

We begin by using only the linear degrees of freedom contained in U to optimise our model and numerical approximations. We shall see that a careful consideration of these linear transformations suggests a natural generalisation of the usual AO basis in a way which enables us to use some of the conclusions of earlier sections where we discussed molecular symmetry. For a given set of non-orthogonal orbitals it is well-known from the work of Lowdin (14) that the simplest solution to equation (34),... 

V = S-1/2 is also the one which is optimum in the sense of generating the closest orthogonal basis possible to the original set. We therefore concentrate on finding optimum non-orthogonal bases and then apply the Lowdin orthogonalisation technique. 

There are numerous ways in which to orthonormalize a basis. Here we choose to employ the symmetric orthonormalization procedure described by Lowdin (51), which has the benefit over other orthogonality procedures that the new basis is as close as possible, in a least squares sense, to the original basis (52)... 

Method/Basis Set Mulliken Lowdin Hirshfeld Bader... 

From a strictly quantum-mechanical point of view, the question of valence-shell expansion and d-orbital participation has two different aspects. One may either look for d-character in Lowdin s natural spin-orbitals of the complete, but unknown, total wave-function or one may ask the question whether the agreement with experimental results is ameliorated significantly when d-orbitals are included in the basis functions of approximate M. 0. calculations. It is a well known tendency for approximate calculations always to ameliorate in certain aspects when the basis is expanded, but also that the extent of this amelioration does not always bear a direct relation to the final results of far more sophisticated calculations. 

There is far less reported experience for ab initio studies of electronically excited states than for ground states. Matrix Hartree-Fock calculations for excited states cannot be considered routine. Often the same basis set is used for both the ground and excited state even though as long ago as 1958 Shull and Lowdin [17] demonstrated... 

Note that we introduced the superscript CD in order to distinguish the expressions obtained by Clark and Davidson from those by Mayer, which will be given in the following marked by Ma. In a similar fashion, Mayer s partitioning of the total spin expectation value can be derived. Starting from Lowdin s expression for the total spin expectation value, Eq. (96), a one-electron basis set is introduced as in Eq. (102) and the numbers of a- and / -electrons, Na and N13, respectively, are replaced by sums over diagonal matrix elements Y (P"S)W and E (P S) w [cf. Eq. (104)], M... 

All electron calculations were carried out with the DFT program suite Turbomole (152,153). The clusters were treated as open-shell systems in the unrestricted Kohn-Sham framework. For the calculations we used the Becke-Perdew exchange-correlation functional dubbed BP86 (154,155) and the hybrid B3LYP functional (156,157). For BP86 we invoked the resolution-of-the-iden-tity (RI) approximation as implemented in Turbomole. For all atoms included in our models we employed Ahlrichs valence triple-C TZVP basis set with polarization functions on all atoms (158). If not noted otherwise, initial guess orbitals were obtained by extended Hiickel theory. Local spin analyses were performed with our local Turbomole version, where either Lowdin (131) or Mulliken (132) pseudo-projection operators were employed. Broken-symmetry determinants were obtained with our restrained optimization tool (136). Pictures of molecular structures were created with Pymol (159). 

Thus the Fock matrix in the X basis is diagonalized by standard methods to yield the MO energies e and the matrix V from which the coefficient matrix c may be obtained by c = uV. There are several ways in which the matrix u and its inverse may be determined. The most commonly used is the symmetric othogonalization due to Lowdin, which involves diagonalization of the overlap matrix. We will not discuss this further. 

To alleviate a number of these problems, Lowdin proposed that population analysis not be carried out until the AO basis functions tp were transformed into an orthonormal set of basis functions / using a symmetric orthogonalization scheme (Lowdin 1970 Cusachs and Politzer 1968)... 

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