Here, /j and rj are the l" left- and the J right-hand eigenvectors of the non-Hermitian Hamiltonian H. The operator is represented on the space spanned by the manifold created by the excitations out of a Hartree-Fock reference determinant, including the null excitation (the reference function). When we calculate the transition probability between a ground state g) and an excited state ]e), we need to evaluate and The reference function is a right-... [Pg.159]

The valence correlation component of TAE is the only one that can rival the SCF component in importance. As is well known by now (and is a logical consequence of the structure of the exact nonrelativistic Bom-Oppenheimer Hamiltonian on one hand, and the use of a Hartree-Fock reference wavefunction on the other hand), molecular correlation energies tend to be dominated by double excitations and disconnected products thereof. Single excitation energies become important only in systems with appreciable nondynamical correlation. Nonetheless, since the number of single-excitation amplitudes is so small compared to the double-excitation amplitudes, there is no point in treating them separately. [Pg.38]

Results for the other open-shell atoms are encouraging. One would expect the P3 method to be considerably less accurate when an unrestricted Hartree-Fock reference state is used. The lowest MAD for B -Ar obtains with the largest of the Dunning sets examined here, i.e. cc-pVQZ. The 6-311++G(3df,3pd) and well-tempered basis sets (WTBS) are roughly equivalent, with MADs of 0.50 eV and 0.57 eV, respectively. [Pg.146]

Errors remain relatively constant for groups III through V, with a sharp increase at group VI. Removal of electrons from (3 spinorbitals in unrestricted Hartree-Fock reference states is relatively poorly described. Absolute errors for the noble gas elements are significantly lower than... [Pg.147]

P3 calculations with unrestricted Hartree-Fock reference states have been reported here for the first time. In addition, a P3 procedure for electron affinities of closed-shell and open-shell systems has been presented. [Pg.155]

Many methods in chemistry for the correlation energy are based on a form of perturbation theory, but the positivity conditions are quite different. Traditional perturbation theory performs accurately for all kinds of two-particle reduced Hamiltonians, which are close enough to a mean-field (Hartree-Fock) reference. There are a myriad of chemical systems, however, where the correlated wave-function (or 2-RDM) is not sufficiently close to a statistical mean field. Different from perturbation theory, the positivity conditions function by increasing the number of extreme two-particle Hamiltonians in which are employed as constraints upon the 2-RDM in Eq. (50) and, hence, they exactly treat a certain convex set of reduced Hamiltonians to all orders of perturbation theory. For the... [Pg.35]

The reconstruction functionals may be understood as substantially renormalized many-body perturbation expansions. When exact lower RDMs are employed in the functionals, contributions from all orders of perturbation theory are contained in the reconstructed RDMs. As mentioned previously, the reconstruction exactly accounts for configurations in which at least one particle is statistically isolated from the others. Since we know the unconnected p-RDM exactly, all of the error arises from our imprecise knowledge of the connected p-RDM. The connected nature of the connected p-RDM will allow us to estimate the size of its error. For a Hamiltonian with no more than two-particle interactions, the connected p-RDM will have its first nonvanishing term in the (p — 1) order of many-body perturbation theory (MBPT) with a Hartree-Fock reference. This assertion may be understood by noticing that the minimum number of pairwise potentials V required to connectp particles completely is (p — 1). It follows from this that as the number of particles p in the reconstmcted RDM increases, the accuracy of the functional approximation improves. The reconstmction formula in Table I for the 2-RDM is equivalent to the Hartree-Fock approximation since it assumes that the two particles are statistically independent. Correlation corrections first appear in the 3-RDM functional, which with A = 0 is correct through first order of MBPT, and the 4-RDM functional with A = 0 is correct through second order of MBPT. [Pg.178]

First consider a Hartree-Fock reference function and transform to the Fermi vacuum (aU occupied orbitals are in the vacuum). Then all particle density matrices are zero and the cumulant decomposition, Eq. (23), based on this reference corresponds to simply neglecting aU three and higher particle-rank operators generated by commutators. This type of operator truncation is used in the canonical diagonalization theory of White [22]. [Pg.357]

We have so far examined the performance of the canonical transformation theory when paired with a suitable multireference wavefunction, such as the CASSCF wavefunction. As we have argued, because the exponential operator describes dynamic correlation, this hybrid approach is the way in which the theory is intended to be used in general bonding situations. However, we can also examine the behavior of the single-reference version of the theory (i.e., using a Hartree-Fock reference). In this way, we can compare in detail with the related... [Pg.375]

Multireference There is no division into occupied and virtual orbitals, all orbitals appear on an equal footing in the ansatz (Equation 8). In particular, the Hartree-Fock reference has no special significance here. For this reason, we expect (and observe) the ansatz to be very well balanced for describing nondynamic correlation in multireference problems (see e.g., refs. 10-12). Conversely, the ansatz is inefficient for describing dynamic correlation, since to treat dynamic correlation one would benefit from the knowledge of which orbitals are in the occupied and virtual spaces. [Pg.152]

The many-body perturbation theory [39] [40] [41] was used to model the electronic structure of the atomic systems studied in this work. The theory developed with respect to a Hartree-Fock reference function constructed from canonical orbitals is employed. This formulation is numerically equivalent to the M ler-Plesset theory[42] [43]. [Pg.286]

M. Urban, P. Neogrady, and I. Hubac, Spin Adaptation in the Open-Shell Coupled-Cluster Theory with a Single Determinant Restricted Hartree-Fock Reference. In R. J. Bartlett (Ed.) Recent Advances in Coupled-Cluster Methods. Recent Advances in Computational Chemistry, Vol. 3. (World Scientific, Singapore, 1997), pp. 275-306. [Pg.41]

Instead of the standard Hartree-Fock reference calculation, a grand-canonical Hartree-Fock calculation [35] is used with the occupation number of a single spin-orbital (i.e., the transition spin-orbital) set to 0.5. Upon convergence, appreciable corrections to the relaxation energy are included in the transition spin-orbital s energy [23, 24], Usually a very close agreement with the ASCF method [36] is obtained [26], The second order electron propagator is applied to the ensemble... [Pg.7]

In this section, we propose to illustrate how the availability of the CRAY has assisted progress in the area of molecular electronic structure. We shall concentrate on two recent advances, namely, the evaluation of the components of the correlation energy which may be associated with higher order excitations, in particular triple-excitations with respect to a single-determinantal, Hartree-Fock reference function, and the construction of the large basis sets which are ultimately going to be necessary to perform calculations of chemical accuracy, that is one millihartree. [Pg.31]

Unlike the second-order and third-order energy diagrams, the fourth-order diagrams can involve intermediate states which are singly-excited, doubly-excited, triply-excited, and quadruply-excited with respect to the Hartree-Fock reference function.8 130... [Pg.24]

Finally, there are seven fourth-order terms which involve an intermediate state which is quadruply-excited with respect to the Hartree-Fock reference function. These diagrams are shown in Figure 8. Time reversal relates diagrams (Bq, Cq), (Dq, Gq), and (Eq, Fq). Explicitly, diagrams Aq, Bq, and Cq, for example, correspond to the expressions... [Pg.25]

Figure 11 Order of perturbation at which various blocks of the configuration mixing matrix first contribute to the energy (a) for Hartree-Fock reference function (b) for ibare-nucleus> reference function (c) for reference function constructed from Brueckner orbitals... [Pg.33]

Bubble Diagrams.—The bubble diagrams which are shown in Figures 13 and 14 are required when the reference function for a closed-shell system is not defined by the matrix Hartree-Fock model, when a restricted Hartree-Fock reference... [Pg.37]

A later study also focused on various means of computing the correlation contribution to the interaction energy in the HE dimer and reached very similar conclusions. All of the correlated methods (MP2, MP4, CCSD(T) and CISD) based on the Hartree-Fock reference configuration gave essentially the same binding energy. The results deteriorate when multireference methods are used. [Pg.76]

Tel. 904-392-1597, fax. 904-392-8722, e-mail aces2 qtp.ufl.edu Ab initio molecular orbital code specializing in the evaluation of the correlation energy using many-body perturbation theory and coupled-cluster theory. Analytic gradients of the energy available at MBPT(2), MBPT(3), MBPT(4), and CC levels for restricted and unrestricted Hartree-Fock reference functions. MBPT(2) and CC gradients. Also available for ROHE reference functions. UNIX workstations. [Pg.416]

P. Neogrady, M. Urban, and I. Hubac, /. Chem. Phys., 100, 3706 (1994). Spin Adapted Restricted Hartree-Fock Reference Coupled-Cluster Theory for Open-Shell Systems. [Pg.126]

X. Li and J. Paldus, ]. Chem. Phys., 102, 2013 (1995). Spin-Adapted Open-Shell State-Selective Coupled-Cluster Approach and Doublet Stability of Its Hartree-Fock Reference. [Pg.127]

Many-Body Perturbation Theory with a Restricted Open-Shell Hartree-Fock Reference. [Pg.131]

D. Jayatilaka and T. J. Lee, Chem. Phys. Lett., 199, 211 (1992). The Form of Spin Orbitals for Open-Shell Restricted Hartree-Fock Reference Functions. [Pg.133]

© 2019 chempedia.info