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Orbital functionals

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. [Pg.392]

Xhe corresponding molecular orbital function for this system is ... [Pg.145]

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. [Pg.125]

Kohn-Sham orbitals functions for describing the electron density in density functional theory calculations... [Pg.365]

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. [Pg.260]

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. [Pg.121]

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... [Pg.131]

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 ... [Pg.85]

Table 6.1. The Spin-Orbit Functions of Different Triplet States That Are Below the Lowest Singlet State [1B3u(n, tt )] of Pyrazineam... 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 ... [Pg.142]

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 ... [Pg.124]

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. [Pg.54]

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... [Pg.149]

General hybrid orbital functions and natural bond angles... [Pg.372]

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,... [Pg.373]

Natural orbital functional vertical ionization potentials obtained from the extended Koopmans theorem. [Pg.419]


See other pages where Orbital functionals is mentioned: [Pg.261]    [Pg.112]    [Pg.117]    [Pg.118]    [Pg.49]    [Pg.187]    [Pg.142]    [Pg.21]    [Pg.257]    [Pg.387]    [Pg.387]    [Pg.389]    [Pg.391]    [Pg.391]    [Pg.393]    [Pg.395]    [Pg.397]    [Pg.399]    [Pg.399]    [Pg.401]    [Pg.403]    [Pg.405]    [Pg.407]    [Pg.409]    [Pg.411]    [Pg.413]    [Pg.415]    [Pg.417]    [Pg.418]    [Pg.418]    [Pg.419]    [Pg.421]    [Pg.422]    [Pg.423]   
See also in sourсe #XX -- [ Pg.680 ]




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Amide functional group atomic orbital structure

Antithesis Orbital Functional Derivatives Define Linear Operators

Atomic orbital basis function

Atomic orbital basis function derivatives

Atomic orbital basis function incomplete

Atomic orbitals functions

Atomic orbitals radial probability function

Atomic orbitals wave functions

Basic Concept of Orbital-Dependent Functionals

Basis functions function Slater-type orbitals

Basis functions numerical orbitals

Bonding orbitals wave function

Canonical molecular orbitals and localized functions

Computational technique atomic orbitals as building blocks of the molecular wave function

Crystal orbitals from Bloch functions (LCAO CO method)

D orbitals, wave functions

D-orbitals functional forms

Density function method Kohn-Sham orbital

Density function theory Kohn-Sham orbitals

Density functional Kohn-Sham orbitals

Density functional theory crystal orbital structures

Density functional theory molecular orbital bonding analysis

Density functional theory orbital-based

Density functional theory orbital-dependent

Density functional theory orbital-free

Density functional theory spin-orbit effects

Density-functional theory and Kohn-Sham orbitals

Detailed configuration, orbital wave functions

Double-zeta Slater functions/orbitals

Eigenvalue natural orbital functionals

Electron configurations orbital wave functions

Exact exchange orbital functionals

Expansion in a basis of orbital wave functions

Explicit construction of the energy density functional within an orbit

Finite basis functions Gaussian-type orbitals

Finite basis functions Slater-type orbitals

Functional derivative, orbital

Gauge-including atomic orbital density functional theory, electron

Gaussian functions orbital energy calculations with

Gaussian functions, spin-orbit operators

Gaussian-type orbital function

Ground-state wave function electronic Hamiltonian, spin-orbit

Group Product Functions and the Basis Orbitals

Highest occupied molecular orbital wave function

Hilbert space, orbital wave functions

Hohenberg-Kohn theorems orbital functional theory

Hybrid orbital nonorthogonality functions

Hydrogen orbitals angular functions

Hydrogen orbitals radial functions

Hydrogen-like orbitals radial distribution function

Independent-electron models orbital functional theory

Kohn-Sham density functional theory, orbital

Kohn-Sham density functional theory, orbital occupation numbers

Kohn-Sham orbitals Density functional theory

Localized functions molecular ‘orbitals

Lowest unoccupied molecular orbital wave function

Many-electron wave functions atomic orbitals approximation

Molecular orbital and density functional

Molecular orbital model, wave function

Molecular orbital theory wave-function coefficients

Molecular orbital wave functions

Molecular orbitals , nuclear magnetic density functional theory, electron

Molecular orbitals basis function expansion

Molecular orbitals from atomic orbital wave functions

Molecular orbitals functions

Molecular-orbital functions

Motivation for Orbital-Dependent Functionals

Natural orbital function

Natural orbital function characterized

Natural orbital function defined

Natural orbital function dipole moment

Natural orbital function energetics

Natural orbital function equilibrium geometries

Natural orbital function practical

Natural orbital function vibrational frequencies

Natural orbital functional

Natural spin-orbital functional

Natural spin-orbital functional characterized

Optimization natural orbital functionals

Orbital angular momentum wave functions

Orbital energy functions

Orbital energy using Slater double-zeta functions

Orbital functional

Orbital functional components

Orbital functional theory

Orbital functional theory of the -matrix

Orbital functionals Schrodinger

Orbital functionals and other nonlocal approximations hybrids, Meta-GGA, SIC, OEP, etc

Orbital functionals correlation

Orbital functionals external potential

Orbital functionals kinetic energy

Orbital radial densities/distribution function

Orbital-Dependent Exchange-Correlation Functional

Orbital-dependent functionals

Orbital-functional response theory, exchange-only

Orbital-specific hybrid functional

Orbitals functions

Orbitals radial distribution function

Orbitals radial probability functions

Orbitals, overview Wave functions

Orbitals, wave-function calculations

Probability functions, orbitals

Radial distribution functions hydrogen atomic orbitals

Radial distribution functions, valence orbitals

Reciprocal orbitals functions

Relativistic molecular orbitals, types functions

Response functions spin-orbit

Self-consistent field method Slater determinant orbital function

Semi-empirical Orbital-Dependent Exchange-Correlation Functionals

Slater functions orbital energy calculations using

Slater type orbitals functions

Slater-type functions, spin orbital products

Spatial function symmetry orbital approximation

Spherical harmonic functions hydrogen atom orbitals

Spin orbitals representation, functionals

Spin-orbit operators functions

Spin-orbit perturbed wave functions

Spin-orbitals orthonormalized functions

Subshell configuration orbital wave functions

Symmetry of Localized Crystalline Orbitals. Wannier Functions

The Pair Density. Orbital-dependent Exchange-correlation Functionals

The Relationship between Molecular Orbital and Valence Bond Wave Functions

The carbon orbitals expanded in HO functions

The carbon orbitals expanded in Laguerre functions

The independent functions from an orbital product

Three-orbital cluster functions

Time-dependent orbital functional theory (TOFT)

Translation and Space Symmetry of Crystalline Orbitals Bloch Functions

Two-Point Approximate Orbital-Free Kinetic Energy Functionals

VB Wave Functions with Localized Atomic Orbitals

Valence Bond Wave Functions with Fragment Orbitals

Variational principles orbital functional theory

Wave function analysis localized molecular orbitals

Wave function analysis natural atomic orbitals

Wave function analysis natural orbitals

Wave function expansions radial orbitals

Wave function for orbital angular momentum

Wave function frozen orbital

Wave function orbital

Wave function orbitals and

Wave functions orbitals

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