Natural orbitals definition

The original definition of natural orbitals was in terms of the density matrix from a full Cl wave function, i.e. the best possible for a given basis set. In that case the natural orbitals have the significance that they provide the fastest convergence. In order to obtain the lowest energy for a Cl expansion using only a limited set of orbitals, the natural orbitals with the largest occupation numbers should be used. 

Although HF orbitals are, by definition, the best possible for a singleconfiguration wavefunction, it is actually possible to find a better set of orbitals, called natural orbitals,31 to describe the correlated p(r). The natural orbitals are maximum-occupancy orbitals, determined from itself and guaranteed to give fastest possible convergence to p(r), i.e., consistently higher occupancies n, than HF orbitals inEq. (1.15). For a HF wavefunction the natural orbitals and HF orbitals are equivalent, but for more accurate wavefunctions the natural orbitals allow us to... 

From this wave function, one sees how even in the early beginning of molecular quantum mechanics, atomic orbitals were used to construct molecular wave functions. This explains why one of the first AIM definitions relied on atomic orbitals. Nowadays, molecular ab initio calculations are usually carried out using basis sets consisting of basis functions that mimic atomic orbitals. Expanding the electron density in the set of natural orbitals and introducing the basis function expansion leads to [15]... 

All this is in contradiction to the density functional definition of electronegativity as presented originally by Parr[36], where it is also inferred that the electronegativity for all the (natural) orbitals of the system should be the same. The discrepancy between these two different derivations and points of view[29,36] is still to be resolved. In addition, the global definition of electronegativity as promulgated within density functional theory, does not appear to be particularly useful, as it yields the same... 

One can argue that for the application of 1) and 2) only trends in orbital nature and orbital populations matter so that any quantum chemical method (HF or DFT) or any way of calculating charges (Mulliken or natural orbital populations) suffices for this purpose as long as it is applied in a consistent way. One can further argue that the silylium cation character of a given silyl ion has only to be determined for the gas phase. Once this has been done, other molecular properties of the ion in question can be calculated and compared with the corresponding measured values for either gas or solution phase. In this way, a purely theoretical definition of the silylium ion character could be adjusted and extended to measurable quantities. 

When r = r, eqn (El.2) becomes eqn (1.11) hence, p(r) is said to be a diagonal element of r< (r, r ). While eqns (1.11) and (El.2) are formally alike, one can calculate the kinetic energy from the latter but not from the former, for only in the latter can one insert the operator between the natural orbitals and let it act separately on or rjf. The average value of a two-electron property can be expressed in terms of the diagonal elements of the second-order density matrix r (ri,r2). Assuming a summation over electron spins, its definition is... 

Another general theorem 9>1( states that the set of NO s associated with every eigenfunction of the Schrodinger equation has definite symmetry properties. In particular, for a planar molecule the natural orbitals are either a or n orbitals. Therefore, a and n orbitals have a physical meaning independent of any model assumption or approximation. By adding the occupation numbers of each species of orbitals, one defines... 

The definition and properties of B(r) may be summarized as follows. For simplicity, we treat the spinless one-electron wave function assuming the independent-particle model or the natural orbital expansion (Lowdin, 1955 Benesch and Smith, 1971). Based on the three-dimensional momentum density p(p),... 

Here, are the so-called natural orbitals, and their occupancies , lie in the interval [0 1]. Per definition, the improvements that are obtained when replacing Eq. (4) by Eq. (11) are the correlation effects. These may be included either through application of the variational principle or perturbatively. 

In the single-determinant case, the MOs are just such a set where, for example, in the closed-shell case the rik of eqn ( 22.1) are all 2 and, in the DODS case, the of eqn ( 22.2) are all 1. There is nothing new here since our choice of a single-determinant of equally-occupied MOs has pre-empted the definition of natural orbitals. But what about the multi-determinant case ... 

The definition of radical given here is sufficient for our discussions, but there are more precise definitions. One is that a radical is "a species with one or more (spin-free) natural orbitals whose occupation numbers are near 1." Klein, D. J. Alexander, S. A. in King, R. B. Rouvray, D. H., Eds. Graph Theory and Topology in Chemistry Elsevier Amsterdam, 1987 p. 404. 

A more rigorous definition of a biradicaloid geometry employs the concept of natural orbitals and their occupancies, which are well defined at all levels of quantum mechanical description of molecules in the Born-Oppenheimer approximation at a biradicaloid geometry, two of the ground state natural orbital occupation munbers are approximately equal to unity (the others, of course, are close to two or close to zero). 

DEFINITION AIM-EN2 The density of a fermionic system can be expressed by its natural orbitals and by the associated occupancy... 

Electronic Diabatic States Definition, Computation, and Applications Localized MO SCF Methods Natural Bond Orbital Methods Natural Orbitals. 

From the definition of p, it follows that the matrix P is hermitian with trace equal to the number of electrons. This quadratic form can be brought to diagonal by defining a new set of natural orbitals... 

All elements, by definition, have a unique proton number, but some also have a unique number of neutrons (at least, in naturally occurring forms) and therefore a unique atomic weight - examples are gold (Au Z = 79, N = 118, giving A =197), bismuth (Bi Z = 83, N = 126, A = 209), and at the lighter end of the scale, fluorine (F Z = 9, N = 10, A = 19) and sodium (Na Z = 11, N= 12, A = 23). Such behavior is, however, rare in the periodic table, where the vast majority of natural stable elements can exist with two or more different neutron numbers in their nucleus. These are termed isotopes. Isotopes of the same element have the same number of protons in their nucleus (and hence orbital electrons, and hence chemical properties), but... 

Practitioners of quantum chemistry employed both the visual imagery of nineteenth-century theoretical chemists like Kekule and Crum Brown and the abstract symbolism of twentieth-century mathematical physicists like Dirac and Schrodinger. Pauling s Nature of the Chemical Bond abounded in pictures of hexagons, tetrahedrons, spheres, and dumbbells. Mulliken s 1948 memoir on the theory of molecular orbitals included a list of 120 entries for symbols and words having exact definitions and usages in the new mathematical language of quantum chemistry. 

Dnring an electron transfer, the acceptor places its LUMO at the electron disposal and the donor releases an electron that is located on its HOMO. These orbitals are frontier orbitals. In the corresponding ion-radicals, the distribution of an unpaired electron proceeds, naturally, under frontier-orbital control. This definitely reflects in the ion-radical reactivity and not always by a self-obvions manner. Let ns concisely trace peculiarities of ion-radical fragmentation reactions that are very important in organic synthesis. 

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