Many-body expansion method

Topographical Aspects, Calculation, and Functional Representation via a Double Many-Body Expansion Method... 

From the numerous studies we have made on potential energy functions using the many body expansion method I pick out the ground state surface of HCO [12] to show the most successfiil use of empirical and ab initio data. The asymptotic limits (two-body terms) to this surface arc provided by the diatomics CH(X, ID,... 

The accurate representation of multidimensional potential energy surfaces is a formidable problem. A common approach is to employ an expression that incorporates as much physical insight as possible in the functional form and which has a number of parameters that are adjusted to empirical data. Examples of this approach are found in applications of the London-Eyring-Polyani-Sato (LEPS), valence-bond (VB), diatomics-in-molecules (DIM), and many-body expansion methods to polyatomic systems. 

Eqs (46) and (49) are the basic equations of the ECC theory described in ref 124. The approximate ECC methods, such as ECCSD, are obtained by truncating the many-body expansions of cluster operators T and Z at some excitation level < N. so that T is replaced by eq (4), and Z is replaced by... 

Finally, we remark that, although the in the present study attention has been focussed on the excited states of some simple atomic systems, the method described here can be applied to molecular systems where they can be expected to support high accuracy as well as providing a suitable reference with respect to which well-founded many-body expansions for correlation effects can be developed. 

There are two issues that have to be addressed before one can use Eqs. (25) or (28) in practical calculations. First of all, the exact MMCC corrections SgA) and < qCCSD, Eqs. (25) and (28), respectively, have the form of long many-body expansions involving all n-tuply excited configurations with n == i/ia + I, ., /V, where N is the number of correlated electrons in a system. Thus, in order to propose the computationally inexpensive MMCC methods, we have to truncate the many-body expansions for SgA> or excitation level This leads to the so-called MMCC( i, mB) schemes [11-15,24,33,34,39,48,120,121], The CR-CCSD(T) and CR-CCSD(TQ) methods [11-14,24,33,34], reviewed and tested in this work, are the MMCC( u, mB) schemes with mA = 2 and mB = 3 (the CR-CCSD(T) case) or 4 (the CR-CCSD(TQ) case). Second of all, the wave function % that enters the exact Eqs. (25) or (28) is a full Cl ground state, which we usually do not know (if we knew the exact ko> state, we would not have to perform any calculations ). Thus, in order to propose the computationally tractable approaches based on the MMCC theory defined by Eqs. (25) and (28), we must approximate fi o) in some way as well. The CR-CCSD(T) and CR-CCSD(TQ) methods employ the low-order MBPT-like expressions to define fi o) [11-14,24,33,34],... 

Although length of the many-body expansion of Hn,open is enormous, very few terms enter a particular approximate scheme. For example, the EOM-CCSD method needs one- and two-body components of H and certain types of the three-body H3 terms [33,34] (for further comments, see Ref. 60 cf., also, Sections 4 and 5). The situation gets complicated if we want to go beyond the EOMCCSD approximation and represent T as a sum of, for example, I), T2, and T3 components. For this reason, the only higher-than-two-body terms included in the current implementation of the EOM-CCSDT scheme are the three-body terms of the EOMCCSD method [46]. This approximation seems to work very well, even though the EOMCCSDT procedure defined in this way remains an n8 method. 

As in the ordinary EOMCC theory, in the EOMXCC method we solve the electronic Schrodinger equation (1) assuming that the excited states are represented by Eq. (7). We use the exponential representation of the ground-state wave function I S o), Eq. (8), but no longer assume that the cluster components Tn result from standard SRCC calculations (see below). The many-body expansions of the excitation operator Rk have the same form as in the ordinary EOMCC formalism. In particular, the three different forms of Rk discussed in the previous section [fi -E, R A, and REqs. (28), (30), and (26), respectively] are used to define the EE-EOMXCC, EA-EOMXCC, and IP-EOMXCC methods. As in the standard EOMCC method, by making suitable choices for the operators Qa, which define Rk, we can always extend the EOMXCC theory to other sectors of the Fock space. 

One can also obtain the Fano formula explicitly from a diagrammatic many-body expansion. There exists a wide variety of alternative theoretical approaches, some of which (e.g. coordinate rotation, projection operator methods, etc) will not even be described in the present monograph. 

Gao, J., and Wang, Y. (2012]. Communication Variational many-body expansion Accounting for exchange repulsion, charge delocalization, and dispersion in the fragment-based explicit polarization method, J. Chem. Phys. 136, 071101. 

---