Orbitals functions

The sign of the last term depends on the parity of the system. Note that in the first and last term (in fact, determinants), the spin-orbit functions alternate, while in all others there are two pairs of adjacent atoms with the same spin functions. We denote the determinants in which the spin functions alternate as the alternant spin functions (ASF), as they turn out to be important reference terms. 

Xhe corresponding molecular orbital function for this system is ... 

In order to obtain this savings in the computational cost, orbitals are symmetry-adapted. As various positive and negative combinations of orbitals are used, there are a number of ways to break down the total wave function. These various orbital functions will obey different sets of symmetry constraints, such as having positive or negative values across a mirror plane of the molecule. These various symmetry sets are called irreducible representations. 

Kohn-Sham orbitals functions for describing the electron density in density functional theory calculations... 

Multiplying a molecular orbital function by a or P will include electron spin as part of the overall electronic wavefunction i /. The product of the molecular orbital and a spin function is defined as a spin orbital, a function of both the electron s location and its spin. Note that these spin orbitals are also orthonormal when the component molecular orbitals are. 

T dtc) and the d Zn(rffc)2, indicate a relatively great stabiUty for electronic states with symmetrical orbital functions. It parallels the maxima in ionisation potentials of the elements with half and completely filled subshells. 

Orbitally degenerate grormd states, in general, cannot be treated in the spin-Hamiltonian approach. In this case, SOC has to be evaluated explicitly on an extended basis of spin-orbit functions. However, in coordination chemistry and bioinorganic chemistry, this is only of marginal importance, because the metal centers of... 

Wave functions for the orbitals of molecules are calculated by linear combinations of all wave functions of all atoms involved. The total number of orbitals remains unaltered, i.e. the total number of contributing atomic orbitals must be equal to the number of molecular orbitals. Furthermore, certain conditions have to be obeyed in the calculation these include linear independence of the molecular orbital functions and normalization. In the following we will designate wave functions of atoms by % and wave functions of molecules by y/. We obtain the wave functions of an H2 molecule by linear combination of the Is functions X and of the two hydrogen atoms ... 

Table 6.1. The Spin-Orbit Functions of Different Triplet States That Are Below the Lowest Singlet State [1B3u(n, tt )] of Pyrazineam...

Triplet state Spin function Spin-orbit function ... 

Since the orbital functions, (0 and mL>l) are orthogonal, the second term vanishes. The absolute value square of the matrix element of a Hermitean operator can be written as ... 

When spin-orbit coupling is introduced the symmetry states in the double group CJ are found from the direct products of the orbital and spin components. Linear combinations of the C"V eigenfunctions are then taken which transform correctly in C when spin is explicitly included, and the space-spin combinations are formed according to Ballhausen (39) so as to be diagonal under the rotation operation Cf. For an odd-electron system the Kramers doublets transform as e ( /2)a, n =1, 3, 5,... whilst for even electron systems the degenerate levels transform as e na, n = 1, 2, 3,. For d1 systems the first term in H naturally vanishes and the orbital functions are at once invested with spin to construct the C functions. 

We have though previously expressed our reservations (10, 11) concerning the combination of ft and ft data for the derivation of a2 and (ZjZ0). Thus, whereas the repulsion parameter, B, is essentially an outer radial quantity (82, 83) the spin-orbit coupling constant, , is dominantly an inner orbital function (84). Moreover the (ZjZo) values derived in most cases indicate a rather small measure of central field covalency. Nevertheless, the a2 values obtained tend to parallel the mm rather than the a 0ot values, especially as the extent of covalency increases from the M(IV) to M(V) to M(VI) series, thus suggesting that flroot values may be too large. On the other hand the values are likely... 

General hybrid orbital functions and natural bond angles... 

We shall now construct two concentrated spd hybrid orbitals that are equivalent, orthonormal, and have maximum concentration of electron density in their respective bonding directions. We can choose any arbitrary direction for the two bond hybrids that we construct. The judicious choice for the direction of the first bond hybrid is along the z axis, because only one of the five d-orbital functions (dz2) can contribute to a bond in this direction. The largest value of the angular part of the first hybrid function (i.e., the maximum concentration of electron density) along the z axis for a chosen spM hybrid orbital hi is insured by choosing both the p and d components to lie along the z direction,... 

Natural orbital functional vertical ionization potentials obtained from the extended Koopmans theorem. 

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