N-electron hierarchy

An important characteristic of ab initio computational methodology is the ability to approach the exact description - that is, the focal point [11] - of the molecular electronic structure in a systematic manner. In the standard approach, approximate wavefunctions are constructed as linear combinations of antisymmetrized products (determinants) of one-electron functions, the molecular orbitals (MOs). The quality of the description then depends on the basis of atomic orbitals (AOs) in terms of which the MOs are expanded (the one-electron space), and on how linear combinations of determinants of these MOs are formed (the n-electron space). Within the one- and n-electron spaces, hierarchies exist of increasing flexibility and accuracy. To understand the requirements for accurate calculations of thermochemical data, we shall in this section consider the one- and n-electron hierarchies in some detail [12]. 

The convergence of the quantum chemical calculations can be studied in terms of two types of hierarchies. First, the quality of the calculations depends on the flexibility of the MO space the AOs that are used to expand the MOs may be extended in a well-defined and systematic manner, thereby establishing a one-electron hierarchy. Second, we can increase the excitation level in coupled-cluster theory or the order of perturbation expansion, thus setting up an n-electron hierarchy of approximate electronic wave functions. In Fig. 5, the roles of the one-electron and the ra-electron hierarchies are illustrated. 

The quality of quantum-chemical calculations depends not only on the chosen n-electron model but also critically on the flexibility of the one-electron basis set in terms of which the MOs are expanded. Obviously, it is possible to choose basis sets in many different ways. For highly accurate, systematic studies of molecular systems, it becomes important to have a well-defined procedure for generating a sequence of basis sets of increasing flexibility. A popular hierarchy of basis functions are the correlation-consistent basis sets of Dunning and coworkers [15-17], We shall use two varieties of these sets the cc-pVXZ (correlation-consistent polarized-valence X-tuple-zeta) and cc-pCVXZ (correlation-consistent polarized core-valence X-tuple-zeta) basis sets see Table 1.1. 

Figure 4. Hierarchy of the SF models. Similar to the non-SF SR methods, the SF models converge to the exact n-electron wavefunction when the spin-flipping operator 0 includes up to n-tuple excitations. For example, the SF-CCSD model...

The exponential ansatz given in Eq. [31] is one of the central equations of coupled cluster theory. The exponentiated cluster operator, T, when applied to the reference determinant, produces a new wavefunction containing cluster functions, each of which correlates the motion of electrons within specific orbitals. If T includes contributions from all possible orbital groupings for the N-electron system (that is, T, T2, . , T ), then the exact wavefunction within the given one-electron basis may be obtained from the reference function. The cluster operators, T , are frequently referred to as excitation operators, since the determinants they produce from fl>o resemble excited states in Hartree-Fock theory. Truncation of the cluster operator at specific substi-tution/excitation levels leads to a hierarchy of coupled cluster techniques (e.g., T = Ti + f 2 CCSD T T + T2 + —> CCSDT, etc., where S, D, and... 

The basis sets described above are small and intended for qualitative or semiquantitative, rather than quantitative, work. They are used mostly for simple wave functions consisting of one or a few Slater determinants such as the Hartree-Fock wave function, as discussed in Sec. 3. For the more advanced wave functions discussed in Sec. 4, it has been proven important to introduce hierarchies of basis sets. New AOs are introduced in a systematic manner, generating not only more accurate Hartree-Fock orbitals but also a suitable orbital space for including more and more Slater determinants in the n-electron expansion. In terms of these basis sets, determinant expansions (Eq. (14)) that systematically approach the exact wave function can be constructed. The atomic natural orbital (ANO) basis sets of Almlof and Taylor [23] were among the first examples of such systematic sequences of basis sets. The ANO sets have later been modified and extended by Widmark et al. [24],... 

The antisymmetric wave functions in the previous section account for electron correlation indirectly through correlation among the coefficients of the geminal or Cl expansions. More compact descriptions of electron correlation are achieved by Jastrow correlation functions that depend explicitly on interelectronic distance. A Jastrow correlation function F can be parameterized in an infinite number of ways. F can be partitioned into a hierarchy of terms. .. TV in which F describes correlations among n electrons. 

The hierarchy of shells, subshells, and orbitals is summarized in Fig. 1.30 and Table 1.3. Each possible combination of the three quantum numbers specifies an individual orbital. For example, an electron in the ground state of a hydrogen atom has the specification n = 1, / = 0, nij = 0. Because 1=0, the ground-state wavefunction is an example of an s-orbital and is denoted Is. Each... 

The Cauchy moments are derived and implemented for the approximate triples model CC3 with the proper N scaling (where N denotes the number of basis functions). The Cauchy moments are calculated for the Ne, Ar, and Kr atoms using the hierarchy of the coupled-cluster models CCS, CC2, CCSD, CC3 and a large correlation-consistent basis sets augmented with diffuse functions. A detailed investigation of the one- and A-electron errors shows that the CC3 results have the accuracy comparable to the experimental results. 

From the A -electron Hilbert-space eigenvalue equation, Eq. (2), follows a hierarchy of p-electron reduced eigenvalue equations [13, 17, 18, 47] for 1 < p < N — 2. The pth equation of this hierarchy couples Dp,Dp+, and and can be expressed as... 

TREE DIAGRAM ILLUSTRATING THE HIERARCHY OF QUANTUM NUMBERS IN THE n = 4 ELECTRON SHELL EACH ORBITAL CAN HOLD TWO ELECTRONS (ONE WITH ms = 1/2 AND ONE WITH ms = -1/2). 

As in the abc hierarchy, each class composing the chemical behavior hierarchy contains only those methods and attributes that pertain to it. Concepts derived at a particular class are used in classes of higher specialization so that more sophisticated concepts can be deduced and discriminating properties elucidated at the proper level. For example, the class POMO contains methods for evaluating the POMO, whether it is the HOMO, the n-HOMO, or any other occupied molecular orbital. These attributes may then be used at more specialized level to expand on the features of an electronic state that gives rise to a particular conceptual behavior. Nucleophilicity illustrates this point, because it may arise from either a negatively charged ion or a neutral atom (i.e., nucleophilicity can... 

TABLE 5. The average cross sections for the halogen atom and ion in the grouml electronic states 4l( P)+,Y+( P) in 10 c n at indicated collision energies e in the laboratory frame of reference for the hierarchy- (6) of interactions and for case a of Hund coui)ling[19, 21] (in parentheses). 

Table 7.2 summarizes the hierarchy among the three quantum numbers. (In Chapter 8, we ll discuss a fourth quantum number that relates to a property of the electron itself.) The total number of orbitals for a given n value is n. ... 

S. K. Rangarajan [1974] A Simplified Approach to Linear Electrochemical Systems. I. The Formahsm n. Phenomenological Couphng HI. The Hierarchy of Special Cases IV. Electron Transfer through Adsorbed Modes,... 

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