Zero-order reference wavefunction

The basic concept behind MBPT is to model the effects of electron correlation by treating them as a perturbation of a zero-order reference wavefunction, Pq- To this end, the correlated wavefunction Tq is expanded in a series, Tq -i- p(i) + p(2) + p(3) +. . . where the superscripts in parentheses indicate the order of the perturbative expansion. Solving the resulting Schrodinger equation yields energies, Eq + -I- -i- > + . ... 

Of course, in the actual computation, one first has to go to all orders to establish the exact Xco, of Eq. (4) from which Eq. (7) follows. Nevertheless, in practice, for atomic and molecular ground states where the shell model holds well and the zero-order reference is a closed-shell, single deferminan-tal HE wavefunction, it remains true that the dominant contribution to the Ecorr comes from the double excitations (electron pair correlations), although for a given state, the exact magnitude of each term of Eq. (4) depends on the computational method and on the function spaces that are used. 

For the multitude of cases in quantum chemistry in which the omnipresence of interelectronic interactions render the use of one-electron models inadequate, simple consideration of the formalism and the meaning of perturbation theory that is based on well-defined zero-order reference wave-functions indicates that not all terms play the same role as regards their contribution to the eigenfunction of each state. Therefore, depending on the problem under consideration, it may be possible, to a good and practical approximation, to partition the total wavefunction in such a way to... 

The first-order MPPT wavefunction can be evaluated in terms of Slater determinants that are excited relative to the SCF reference function k. Realizing again that the perturbation coupling matrix elements I>k H i> are non-zero only for doubly excited CSF s, and denoting such doubly excited i by a,b m,n the first-order... 

Note that f is at most a two-particle operator and that T is at least a one-particle excitation operator. Then, assuming that the reference wavefunction is a single determinant constructed from a set of one-electron functions. Slater s rules state that matrix elements of the Hamiltonian between determinants that differ by more than two orbitals are zero. Thus, the fourth term on the left-hand side of Eq. [48] contains, at the least, threefold excitations, and, as a result, that matrix element (and all higher order elements) necessarily vanish. The energy equation then simplifies to... 

For molecules, especially in excited states, the choice of the proper, for each problem, extended set of zero-order orbitals and corresponding configurations that would allow, to a good approximation, the recognition, in quantitative terms, of the main features of the wavefunction and the bonds constitutes a challenging problem. For example, such a problem is discussed in Sections 9 and 10, where the exceptional bond of Be2 X E+ is examined in the framework of ND versus D correlation using as reference points the Fermi-seas of the low-lying states of Be. 

Figure 2.1 A hierarchy of CASSCF calculations that reveal the critical significance of the presence of d orbitals in the zero order, Fermi-sea set of orbitals. The curve for CASSCF (4, 26)/+3s, +3d, +3p is in essential agreement with that of the MRCISD, whose reference wavefunction is the standard CASSCF (4, 8)/2s -1- 2p. (The aug-cc-pVDZ basis set was used).

Finally, the last PEC represents = ao T + and is obtained from a MRCISD calculation based on the CASSCF (4, 8)/2s + 2p zero-order wave-function. In such a calculation, the corrections that are taken into account by are considered part of the D correlation. The difference befween the last two curves is very small. Obviously, when additional correlations of the D type (in terms of configurations and basis sets) are included in a full Cl compulation, the inner minimum goes down close to its true value while the outer one disappears [95,103,115]. The same occurs in the present work using the larger CASSCF (4, 26) reference wavefunction with the Be Fermi-sea of (2s, 2p, 3s, 3p, 3d] orbitals. 

In multireference perturbation theory, defining a proper zero-order Hamiltonian is anything but straightforward. The reference wavefunction, in general, is not an eigenfunction of the zero-order Hamiltonian. A second complication arises as interactions between the FOIS functions and zero-order wavefunction through the zero-order Hamiltonian cannot be excluded. Therefore, projection techniques are commonly employed. In NEVPT2, the zero-order Hamiltonian takes the form... 

CASPT2 is most useful for calculations on excited states and diradicals, where multireference wavefunctions are required. However, there are methods available for including electron correlation for radicals and radical ions for which single-determinantal wavefunctions represent good zero-order approximations, without resorting to multideterminantal (i.e., CASSCF) reference wavefunctions. Two of these methods are discussed in the following sections, and we recommend them over CASPT2 for most calculations on molecules with just one unpaired electron. 

When the Hartree-Fock reference determinant offers a very good zero-order description of the system, and no individual determinants or group of determinants make large contributions to the correlated many-body wave-function, the molecule is said to be dominated by dynamical correlation. In such a scenario, the cluster expansion of the wavefunction converges rapidly, and a gold standard correlation method like CCSD(T) [1] truly comes very close to the exact (i.e., full Cl) basis set correlation energy. 

However, before going into a detailed discussion of various relativistic Hamiltonians we will introduce an alternative form of the electronic Hamiltonian (3.4), which is useful for wavefunction-based correlation methods. It is obtained by switching to a particle-hole formalism and then introducing normal ordering. In the second-quantization formalism creation and annihilation operators refer to some specific set of (orthononnal) orbitals, and Slater determinants in Hilbert space translate into occupation-number veetors in Fock space. The annihilation operators in equation 3.4 by definition give zero when acting on the vacuum state... 

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