The Natural Orbitals

A Hartree-Fock calculation produces a set of molecular orbitals, known as the canonical Hartree-Fock orbitals. These orbitals are the solution to the Hartree-Fock-Roothaan equations and, while they may be used as the basis for a Cl expansion of the exact wave-function, they are not optimal in the sense that a different choice of orbital basis may result in a Cl expansion that converges more rapidly to the FCI limit. In 1955, Ldwdin demonstrated that the optimal one-electron basis for the Cl expansion of the exact wavefunction is the natural orbital basis [18]. In order to obtain the natural orbitals, we must first construct the first order reduced density matrix (RDM), defined as  

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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... 

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. 

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. 

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. 

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 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. 

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. 

The quality of the ) states has been tested through their energy and also their transition moment. Moreover from the natural orbitals and Mulliken populations analysis, we have determined the predominant electronic configuration of each ) state and its Rydberg character. Such an analysis is particularly interesting since it explains the contribution of each ) to the calculation of the static or dynamic polarizability it allows a better understanding in the case of the CO molecule the difficulty of the calculation and the wide range of published values for the parallel component while the computation of the perpendicular component is easier. In effect in the case of CO ... 

In the case ofthe VB wave function, the natural orbitals and occupation numbers obtained by diagonalizing the matrix representation of the VB one-density function on the atomic basis set were used in Equation (14). 

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... 

As shown by P.-O. Lowdin,40 the complete information content of y can be obtained from its eigenorbitals, the natural orbitals 9t, and the corresponding eigenvalues n,... 

To And the natural orbitals 0 of this system, we diagonalize the 2 x 2 matrix y(AO) to obtain the eigenvectors and eigenvalues shown below ... 

From the eigenvectors we obtain the natural orbitals in the form... 

The natural orbital 0i is equivalent to the variational Hartree-Fock Is orbital in this case, much closer to the exact hydrogenic solution discussed in Section 1.2. 

The natural orbital analysis indicated that the major configuration for both JT states is ls ) near the minimum [86], While this configuration remains dominant for the state throughout the whole range of R, the state... 

The choice of the active spaces for the MCSCF calculations on HF was based on the natural orbital occupancy numbers obtained in MP2 calculations... 

An important result, found for the SDTQ[N/N] wavefiinctions of all molecules considered, is that the split-localized molecular orbitals yield a considerably faster convergence for truncated expansions than the natural orbitals. For example, for NCCN SDTQ[18/18], millihartree accuracy is achieved by about 50,000 determinants of the ordering based on split-localized orbitals whereas about 150,000 determinants are needed for the natural-orbital-based ordering. This observation calls for the revision of a widely held bias in favor of natural orbitals. 

Additional insight may be obtained by writing the system of equations in the natural-orbital basis set, that is, the basis set that diagonalizes the 1-RDM. In this basis set the two terms with the connected 3-RDM may be collected to obtain the formula for the elements of the connected (or cumulant) 3-RDM [26],... 

The major advantage of a 1-RDM formulation is that the kinetic energy is explicitly defined and does not require the construction of a functional. The unknown functional in a D-based theory only needs to incorporate electron correlation. It does not rely on the concept of a fictitious noninteracting system. Consequently, the scheme is not expected to suffer from the above mentioned limitations of KS methods. In fact, the correlation energy in 1-RDM theory scales homogeneously in contrast to the scaling properties of the correlation term in DPT [14]. Moreover, the 1-RDM completely determines the natural orbitals (NOs) and their occupation numbers (ONs). Accordingly, the functional incorporates fractional ONs in a natural way, which should provide a correct description of both dynamical and nondynamical correlation. 

Based on the 1,3-contracted Schrodinger equation (CSE), the AGP energies for various systems have been recalculated with considerable success [23]. Starting from a geminal g, the 2- and 3-RDMs of the respective AGP wavefunction were used in the evaluation of the 1,3-CSE for the determination of a 1-RDM. From this 1-RDM the natural orbitals 4>i and occupation numbers <, p of the next geminal g = ii 4>2i-i4>2i were derived. Then WQ-D AGp[g]]) was optimized... 

The 1-matrix can be diagonalized and its eigenfunctions are the natural orbitals. Equation (41) then implies that the natural orbital occupation numbers he between zero and one, inclusive. Except for the normalization condition. 

Since the overwhelming majority of wavefunctions are constructed from a one-particle basis set Xjif), j = 1,..., n, the natural orbitals can be expressed in that basis set as... 

In atoms, the natural orbitals can be written as products of radial functions and spherical harmonics... 

The concept of the molecular orbital is, however, not restricted to the Hartree-Fock model. Sets of orbitals can also be constructed for more complex wave functions, which include correlation effects. They can be used to obtain insight into the detailed features of the electron structure. One choice of orbitals are the natural orbitals, which are obtained by diagonalizing the spinless first-order reduced density matrix. The occupation numbers (T ) of the natural orbitals are not restricted to 2, 1, or 0. Instead they fulfill the condition ... 

For systems with more than one electron pair, the simple picture illustrated above obviously breaks down. The approximate validity of the independent electron-pair model, however, still makes it possible to estimate different correlation effects also in many-electron systems from an inspection of the natural orbital occupation numbers. 

The super-CI method now implies solving the corresponding secular problem and using tpq as the exponential parameters for the orbital rotations. Alternatively we can construct the first order density matrix corresponding to the wave function (4 55), diagonalize it, and use the natural orbitals as the new trial orbitals in I0>. Both methods incorporate the effects of lpq> into I0> to second order in tpq. We can therefore expect tpq to decrease in the next iteration. At convergence all t will vanish, which is equivalent to the condition ... 

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