Orbitals natural

The natural spin-orbitals a,(l) corresponding to a many-electron wave function are defined to be the set in terms of which the first-order density matrix is diagonal [36-52], Thus, if we solve the secular equation 

In terms of the a s, the first-order density operator p, (1,1 ) of equation (33) can be written in the form 

In the case where is a single Slater determinant, the eigenvalues n, of the first-order density matrix reduce to occupation numbers, with values either 0 or 1. In the more general case, where is a sum of Slater determinants, the eigenvalues rii can be shown to obey the relationships 

To illustrate this method, we have calculated the natural orbitals of the ground state of lithium (Fig. 1). The basis of one-electron orthonormal spin-orbitals Pa used to represent the first-order density matrix consisted of 25 spin-up and 25 spin-down orthogonalized Coulomb Sturmians. The first-order density matrix flius constructed was block-diagonal. The eigenvalues (occupation numbers) corresponding to the spin-up block were 

As expected, the two spin-up natural orbitals with occupation numbers close to 1 were found to resemble Is and 2s orbitals, respectively. In Fig. 1, the 2s-like natural orbital can be seen to have a radial distribution function with a maximum at r = 3.2 bohrs, while the radial distribution function of the Is-like natural orbital has almost no amplitude beyond r = 2 bohrs. The Is-like spin-down natural orbital also has an occupation number near to 1, 

Properties can be computed by finding the expectation value of the property operator with the natural orbitals weighted by the occupation number of each orbital. This is a much faster way to compute properties than trying to use the expectation value of a multiple-determinant wave function. Natural orbitals are not equivalent to HF or Kohn-Sham orbitals, although the same symmetry properties are present. 

However, this convergence is not monotonic. Sometimes, the smallest calculation gives a very accurate result for a given property. There are four sources of error in ah initio calculations  

The disadvantage of ah initio methods is that they are expensive. These methods often take enormous amounts of computer CPU time, memory, and disk space. The HF method scales as N, where N is the number of basis functions. This means that a calculation twice as big takes 16 times as long (2 ) to complete. Correlated calculations often scale much worse than this. In practice, extremely accurate solutions are only obtainable when the molecule contains a dozen electrons or less. However, results with an accuracy rivaling that of many experimental techniques can be obtained for moderate-size organic molecules. The minimally correlated methods, such as MP2 and GVB, are often used when correlation is important to the description of large molecules. 

Jensen, Introduction to Computational Chemistry John Wiley Sons, New York (1999). 

Veszpremi, M. Feher, Quantum Chemistry Fundamentals to Applications Kluwer, Dordrecht (1999). 

This is a process which increases as the third power of the size of the basis set, and the optimization of the function is therefore an method. The Edmiston- 

Although the localization by energy criteria (Edmiston-Ruedenberg) may be considered more fundamental than one based on distance (Boys) or atomic charge (Pipek-Mezey), the difference in computational effort means that the Boys or Pipek-Mezey procedures are often used in practice, especially since there is normally little difference in the shape of the final LMOs. 

The electron density calculated from a wave function is given as the square of the function, The reduced density matrix of order y, is defined as  

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. 

When natural orbitals are determined from a wave function which only includes a limited amount of electron correlation (i.e. not full Cl), the convergence property is not rigorously guaranteed, but since most practical methods recover 80-90% of the total electron correlation, the occupation numbers provide a good guideline for how important a given orbital is. This is the reason why natural orbitals are often used for evaluating which orbitals should be included in an MCSCF wave function (Section 4.6). 

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Atomic natural orbital (ANO) basis sets [44] are fonned by contracting Gaussian fiinctions so as to reproduce the natural orbitals obtained from correlated (usually using a configuration interaction with... 

In the derivation used here, it is clear that two approximations have been made—the configurations are incoherent, and the nuclear functions remain localized. Without these approximations, the wave function fonn Eq. (C.l) could be an exact solution of the Schrddinger equation, as it is in 2D MCTDH form (in fact is in what is termed a natural orbital form as only diagonal configurations are included [20]). 

This pieture is that deseribed by the BO approximation. Of eourse, one should expeet large eorreetions to sueh a model for eleetronie states in whieh loosely held eleetrons exist. For example, in moleeular Rydberg states and in anions, where the outer valenee eleetrons are bound by a fraetion of an eleetron volt, the natural orbit frequeneies of these eleetrons are not mueh faster (if at all) than vibrational frequeneies. In sueh eases, signifieant breakdown of the BO pieture is to be expeeted. 

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

There are several types of basis functions listed below. Over the past several decades, most basis sets have been optimized to describe individual atoms at the EIF level of theory. These basis sets work very well, although not optimally, for other types of calculations. The atomic natural orbital, ANO, basis sets use primitive exponents from older EIF basis sets with coefficients obtained from the natural orbitals of correlated atom calculations to give a basis that is a bit better for correlated calculations. The correlation-consistent basis sets have been completely optimized for use with correlated calculations. Compared to ANO basis sets, correlation consistent sets give a comparable accuracy with significantly fewer primitives and thus require less CPU time. 

The CBS models use the known asymptotic convergence of pair natural orbital expansions to extrapolate from calculations using a finite basis set to the estimated complete basis set limit. See Appendix A for more details on this technique. 

For aP electron pairs, the coefficient matrix C may be diagonalized, yielding the pair natural orbital (PNO) expansion of the pair energies ... 

In natural orbital form, the asymptotic convergence of has been shown to have the following form, resulting in the CBS limit, ef fCBS) ... 

The filled and hollow circles indicate the contributions of each successive natural orbital. The filled circles correspond to complete shells. Only these points are useful for extrapolating to the complete basis set limit. 

Table 4.4 Natural orbital occupation numbers for the distorted acetylene model in Figure 4.11. Only the occupation numbers for the six central orbitals are shown...

One popular way of obtaining general contraction coefficients is from Atomic Natural Orbitals (ANO), to be discussed in section 5.4.4. The difference between segmented and general contraction may be illustrated as shown in Figure 5.2. 

A MP2/6-311- -G(2df,2p) calculation is carried out, which automaticaUy yields the corresponding HF energy. The MP2 result is extrapolated to the basis set limit by the pair natural orbital method. 

Anotiier way of justifying the use of eq. (6.4) for calculating tire kinetic energy is by reference to natural orbitals (eigenvectors of the density matrix. Section 9.5). The exact kinetic energy can be calculated from the natural orbitals (NO) arising from tire exact density matrix. 

The orbital occupation numbers n, (eigenvalues of the density matrix) will be between 0 and 1, corresponding to the number of electrons in the orbital. Note that the representation of the exact density normally will require an infinite number of natural orbitals. The first N occupation numbers N being the total number of electrons in the system) will noraially be close to 1, and tire remaining close to 0. 

The concept of natural orbitals may be used for distributing electrons into atomic and molecular orbitals, and thereby for deriving atomic charges and molecular bonds. The idea in the Natural Atomic Orbital (NAO) and Natural Bond Orbital (NBO) analysis developed by F. Weinholt and co-workers " is to use the one-electron density matrix for defining the shape of the atomic orbitals in the molecular environment, and derive molecular bonds from electron density between atoms. 

Because of the success of the r12 method in the applications, one had almost universally in the literature adopted the idea of the necessity of introducing the interelectronic distances r j explicitly in the total wave function (see, e.g., Coulson 1938). It was there-fore essential for the development that Slater,39 Boys, and some other authors at about 1950 started emphasizing the fact that a wave function of any desired accuracy could be obtained by superposition of configurations, i.e., by summing a series of Slater determinants (Eq. 11.38) built up from a complete basic one-electron set. Numerical applications on atoms and molecules were started by means of the new modern electronic computers, and the results have been very encouraging. It is true that a wave function delivered by the machine may be the sum of a very large number of determinants, but the result may afterwards be mathematically simplified and physically interpreted by means of natural orbitals.22,17... 

The natural orbitals %2v and %3p are, in contrast to the hydrogenlike functions, localized within approximately the same region around the nucleus as the Is orbital. This means that the polarization caused by the long-range interaction is associated mainly with an angular deformation of the electronic cloud on each atom. If %2p and %3p are expanded in the standard hydrogen-like functions, an appreciable contribution will again come from the continuum. 

Lowdin, P.-O., and Shull, H., Phys. Rev. 101, 1730, Natural orbitals in the quantum theory of two-electron systems/ ... 

An appealing way to apply the constraint expressed in Eq. (3.14) is to make connection with Natural Orbitals (31), in particular, to express p as a functional of the occupation numbers, n, and Natural General Spin Ckbitals (NGSO s), yr,, of the First Order Reduced Density Operator (FORDO) associated with the N-particle state appearing in the energy expression Eq. (3.8). In order to introduce the variables n and yr, in a well-defined manner, the... 

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