Orbitals 296 INDEX

The first summation requires electron repulsion integrals with four virtuaJ indices. Efficient algorithms that avoid the storage of these integrals have been discussed in detail [20]. For every orbital index, p, this OV contraction must be repeated for each energy considered in the pole search it is usually the computational bottleneck. 

R.L. Lopez de Compadre et al., The role of hydrophobicity in the Ames test. The correlation of the mutagenicity of nitropolycyclic hydrocarbons with partition coefficients and molecular orbital indexes. Ini. J. Quantum Chem. 34, 91-101 (1988). 

This defines a Gateaux functional derivative [26, 102], whose value depends on a direction in the function space, reducing to a Frechet derivative only if all e, are equal. Defining Tt = t + v, an explicit orbital index is not needed if Eq. (5.10) is interpreted to define a linear operator acting on orbital wave functions, TL — v = t. The elementary chain rule is valid when the functional... 

This implies that for each special case of orbital index combinations, an orbital gradient ... 

Upon choosing an active orbital index pair i,j) and a rotation sine, variation of the energy through a EJR can be obtained from the preceding section. The net result is a fourth order sine polynomial... 

In 1984, Adamowicz and Bartlett used a linear combination simple spherical Gaussians (FSGOs) for constructing correlation orbitals. They performed SCF calculation with extended basis sets of M functions, which give resulting Woccupied (index a, h,. .. ) and M-N virtual orbitals (index i,j,. . . ). Then, the second order perturbation energy can be expressed as. 

This completes the loop over the virtual orbital index b (ib). 

A single dot on an orbital index, m, indicates restriction to inactive orbitals, and a double dot, m, indicates restriction to active orbitals. 

For the more restrictive CSF expansion spaces, such as PPMC and RCI expansions, it occurs that entire transition density vectors will vanish for particular combinations of orbital indices. It is most convenient if a logical flag is set during the construction of the three vectors, one flag for each vector, to indicate that it contains non-zero elements. This avoids the eflbrt required to check each element individually for these zero vectors. The updates of the elements of the matrix C that result from a particular transition density vector involve two DO loops one over the CSF index, which determines the second subscript of the matrix C, and the other over an orbital index, which, combined with a density matrix orbital index, is used to determine the first subscript of C. Either choice of the ordering of these two loops results in an outer product matrix assembly method. 

This is a Gateaux derivative, whose value varies with direction in the function space [1,2]. An explicit orbital index is not needed if 87 /8p is considered to be the linear operator 77 — v = t. This confirms the chain rule for functional derivatives... 

Tuppurainen K, Lotjonen S, Laatikainen R, Vartiainen T, Maran U, Strandberg M, Tamm T. About the mutagenicity of chlorine-substituted furanones and halopropenals. A QSAR study using molecular orbital indexes. Mutat Res 1991 247 97-102. 

Another useful notion is the so-called generalized bond index Kab referred to the given atoms A and B. According to [36], in case of CIS excited states, Kab, more exactly, Kab[C1S], turns out to be connected with CT numbers Ia b and some additional quantities. Before giving expressions applicable also to RPA and related models, we briefiy clarify the meaning of the generalized bond index. Even within the elementary MO theory, such as the tt-electron model, the corresponding quantity, that is /sT,uy[MO], is not the same as the usual bond order Instead, the squared bond order makes its appearance, namely, by definition the orbital index... 

A state or orbital index has been dropped for the sake of clarity. With V = 0 this equation reduces to the free-particle Dirac equation of chapter 5. For V = Ir we have the equation for Dirac hydrogen-like atoms for point-... 

The two spin densities add up to the total charge density, p r,t) = p t,i) + Pl r,t), and each spin density is written as the sum over the probability densities of the molecular orbitals (where i is an orbital index and [Pg.136]

The case of a molecule (or an extended system with no translational symmetry) is a special case of this formalism when one does not use a cell index, only an orbital index. The form of all the equations remains in this case the same, except that one must omit the k-dependence, and in the definition of the charge-bond-order matrix elements [see equation... 

From now on, we will use the simplified notations (pi) = 10 and 10 ) = ia), in other words the subscript a on the orbital index indicates that it is an oscillator orbital. We designate the occupied (canonical or localized) molecular orbitals as i,j, k,... and the canonical virtual molecular orbitals (VMOs) as a, b,c,. 

If any two spin-orbitals are the same the projected function simply vanishes. This vanishing is the basis of what is usually called the Pauli exclusion principle. The function (O Eq. 2.131) is clearly a determinant of spin-orbitals with the spin-orbital index designating a row (column) and the electron numbering designating a column (row). This was first recognized by Slater and so such determinants are called Slater determinants and often denoted by the shorthand... 

Here we have used abbreviations io for the orbital index and ib for the basis member index. Let us assume we can write our problem as HC = ESC and focus on just one typical matrix element using inner-index sums for the multiplication steps using row-column multiplication. 

The corrections to the expansion coefficients (11.9.14) arising from the replacement of orbital k as described by (11.9.10) may thus be obtained by carrying out a direct Cl iteration (i.e. contraction) involving a one-electron operator with one fixed orbital index. It should be noted that, although... 

---