CCD wave functions

Since the excitation operators (13.1.2) commute among one another, there are no problems with the order of the operators in this expression. The resulting wave function (13.1.5) corresponds to a particularly simple realization of the coupled-cluster model, in which only double excitations are allowed the coupled-cluster doubles (CCD) wave function. For a complete specification of this model, we must also describe the method by which the amplitudes are determined. We shall... 

To investigate whether this is due to limitations of the CCD wave function, or the way in which the CCD energy and amplitudes are conventionally obtained, we have recently performed variational CCD calculations (14). Variational CCD calculations are a restricted form of full configuration interaction (with factorial cost), because the energy expectation value includes contributions from all orbital replacements as high as the number of electrons ... 

So, restricting ourselves to the 1 1 perfect pairing active space, the simplest possible version of a CCD wave function would be to retain only the linear number of excitations needed to provide alpha-beta bond-antibond correlations ... 

Perhaps the next simplest form of trial CCD wave function would be to retain the 1 1 active space, but reintroduce some coupling between valence electron that are associated with different pairs. The simplest way to do this defines what we have termed the imperfect pairing (IP) operator (24) ... 

To this stage we have established that the simple CCD wave function is capable of yielding good accuracy when applied in a valence space, with an... 

The micrograph or the image obtained on an EM screen, photographic film, or (more commonly today) a CCD is the result of two processes the interaction of the incident electron wave function with the crystal potential and the interaction of this resulting wave function with the EM parameters which incorporate lens aberrations. In the wave theory of electrons, during the propagation of electrons through the sample, the incident wave function is modulated by its interaction with the sample, and the structural information is transferred to the wave function, which is then further modified by the transfer function of the EM. 

Various approximations of Eq. (6), the exact wave function in the coupled-cluster formalism, have been discussed in the chemical literature. In particular, Cizek s coupled-pair many-electron theory (CPMET),5 also referred to as coupled-cluster doubles (CCD) by Bartlett,8 has re-... 

The Brueckner orbital variant of CC should also be mentioned. CCSD puts in all single excitation effects via the wavefunction exp(Ti + T2) o- We can instead change the orbitals ip, in Oo in this wavefunction until Tj = 0. These orbitals are called Brueckner orbitals and define a single determinant reference B instead of o that has maximum overlap with the correlated wave-function. Since B-CCD " (or BD) effectively puts in Tj, it will give results similar but not identical to those from CCSD (they differ in fifth order). For BH, the corresponding B-CCD errors are 1.81, 2.88, and 5.55, compared to 1.79, 2.64 and 5.05, for CCSD as a function of R. . See also B-CCD for symmetry breaking problems. ... 

In the 1960s and early 1970s, even the formalism of second quantization was shunned by most quantum chemists, even deemed as an unnecessary extravagance. For this reason, we wrote a paper [85] in which we derived the CCD equations using the standard first quantization wave-function formahsm, without any diagrams (for a similar CCSD version, see Ref. [86], written in connection with the appearance of the quadratic Cl [87]). Yet, even here, we tried for too much brevity and compact mathematical notation. For example, the key expression for the disconnected quadruples in terms of doubles, later on usually written in its full form listing all 18 terms (see, e.g. the last Eq. (33) of Ref. [88] note already a much more compact form used in Eq. (11) of Ref. [89]),... 

Inclusion of only J2 gives an approximate CC approach called the coupled-cluster doubles (CCD) method. Since = + T2 + lf + , the wave function i ccd contains determinants with double substitutions, quadruple substitutions, hextuple substitutions, and so on. Recall (Section 15.17) that quadruple substitutions are next in importance after double substitutions in a Cl wave function. The treatment of quadruple substitutions in the CCD method is only approximate. The CCD quadruple excitations are produced by the operator tl, and so the coefficients of the quadruply substituted determinants are determined as products of the coefficients of the doubly substituted determinants [see Eq. (15.92)], rather than being determined independently, as in the CI-SDTQ method The CCD approximation of the coefficients of the quadruply substituted determinants turns out to be pretty accurate. 

X) X = CCD, ECCD) is the expectation value of the Hamiltonian with the wave function, where T2 is obtained with method X (cf. Eq. (119)). 

The substantial improvement in the description of the S4 model system offered by the ECCD approach suggests that the T2 clusters resulting from the ECCD calculations are considerably better than the CCD T2 values. This can be seen by calculating the expectation values of the Hamiltonian, designated by (X), where X = CCD or ECCD, with the normalized CCD-like wave functions... 

The second modern geminals-based approach was developed by Szalewicz and co-workers in the early 1980s. These authors are mostly interested in the closed-shell CCD method, its linearized version (LCCD), and the second-(MP2) and third-order (MP3) Mpller-Plesset perturbation theory approaches, which are contained in LCCD, In single-reference CCD theory, the wave function is written in the exponential form... 

In the following, the theory of Kutzelnigg s linear R12 functions shall be presented and analyzed in the framework of the coupled-cluster doubles (CCD) method, To illustrate the ideas and approximations employed in the linear R12 methods, it is sufficient to consider the CCD model, as the corresponding CCD-R12 theory exhibits all properties of the R12 theories. It is a relatively simple matter to include singles (CCSD) or even triples (for example in the CCSD(T) method), and CI-R12-type wave functions or MPn-R12 energies require essentially the same computational procedures as the CCD-R12 approach. 

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