Natural orbital function defined

It turns out that a Cl calculation using natural orbitals converges much faster than one using SCF orbitals, so one needs to include far fewer CSFs to obtain a given level of accuracy. Unfortunately, the natural orbitals are defined in terms of the final Cl wave function and cannot be calculated until a Cl calculation using SCF orbitals has been completed. Hence several schemes have been devised to calculate approximate natural orbitals and use these in Cl calculations. 

Natural orbitals were defined by P. O. Lbwdin as the eigenfunctions of the first-order reduced density matrix. They provide a simple factorization of wave functions for two-electron systems which brings them into a standard, easily interpreted form. For many-electron systems, they provide a basis for constructing Slater determinants so that the importance of the first few terms is maximized. Used in this way, they were the basis for the construction of some of the first accurate wave functions for molecules. With modern computers, the number of Slater determinants involved in the wave function is no longer so much of an issue and natural orbitals are now mainly used to reduce the wave function and density matrix to a reasonably compact form that facilitates interpretation. 

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... 

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. 

Transformations. A frequently occurring step in calculations is a change of basis via a linear transformation. That is, a new set of basis functions (such as molecular orbitals, group orbitals, natural orbitals, etc.) are defined as linear combinations of the original atomic orbitals, by... 

Kraka and Cremer have estimated the biradical character of the benzynes in two ways. The first involves the occupation number of the natural orbitals ( j) obtained from the CCSD(T) wavefunction. The biradical character is then defined as S , - S j(reference) where the reference here is benzene. With this method, the biradical character is 11 percent, 20 percent, and 65 percent for 41, 42, and 43, respectively. Alternatively, one can use the largest amplitude for a doubles excitation within the CCSD(T) wavefunction to indicate how strongly the next most important configuration (after the HF configuration) contributes to the total wave-function. This ampUrnde is 0.24 for 41, 0.34 for 42, and 0.71 for 43. Yet another measure of biradical character is the ratio where these are the coeffi-... 

The fact is that if a given function of n (vector) variables is to be approximated by a sum of products (or determinants if the function is antisymmetric) of functions of a single (vector) variable then this approximation is at its numerical best for a given length of expansion (AT in eqn ( 22.3)) when the single-variable functions are the natural orbitals defined above. Conversely, for a given least-squares accuracy, the number of terms in the expansion eqn ( 22.3) is at its minimum when the single-particle orbitals of which the determinants are composed are natural orbitals. 

In order to define natural orbitals, we now consider the first-order reduced density matrix of an iV-electron system. Given a normalized wave function, O, then 0(xi,..., x y), Xjy) dx dxjy is the probability... 

This shows that p is similar to the well-known one-particle density of electronic structure theory. Diagonalizing the operator p yields the natural populations and the natural orbitals" defined as the eigenvalues and eigenvectors of p(p). Since we are dealing with distinguishable particles we have a separate density matrix for each degree of freedom. The natural populations characterize the contribution of the related natural orbital to the MCTDH wave function. Small natural populations therefore indicate that the MCTDH expansion converges. The natural populations thus provide us with... 

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 ... 

Meanwhile orbitals cannot be observed either directly, indirectly since they have no physical reality contrary to the recent claims in Nature magazine and other journals to the effect that some d orbitals in copper oxide had been directly imaged (Scerri, 2000). Orbitals as used in ab initio calculations are mathematical figments that exist, if anything, in a multi-dimensional Hilbert space.19 Electron density is altogether different since it is a well-defined observable and exists in real three-dimensional space, a feature which some theorists point to as a virtue of density functional methods. 

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