CI energy

Cycloadduct R Total cis Energy trans kcal/mol TE cis/trans (Exp)... 

In column A, use is made of the variational MRSD-CI energies. In column B, these energies are eorrected for higher excitations (see text)... 

A seventh degree polynomial fit to the theoretical curves has been used (columns A and B). In column A use is made of the variational MR SD-Cl energies while in column B estimated full-CI energies 111] are utilised... 

Table 7.1 Basis set convergence for HF and full CI energies of CO and O, respectively...

Another feature deals with the trans, trans and trans, cis conformations of DPC, as shown in Fig. 55. The calculations indicate that the trans, trans conformation of DPC is preferred by 4.7 kj mol 1 over the trans, cis conformation, with a barrier of 4.7 kj mol 1 between these two conformations. The trans, cis energy minimum is at a torsion angle of 150° about the Cc - O" axis relative to the trans, trans geometry. 

Figure 15-5. The potential-energy profiles of 5-fluorocytosine (left) and 6-fluorocytosine (right). For the 5-fluorocytosine, the open circles represent the CIS energies, and the open squares denote the CCSD(T) energies of the 1 -tt-tt state shifted by 1.45455 au. For the 6-fluorocytosine, the circles (CIS) and the squares (CCSD(T)) are the 1 -tt-tt state energies, whereas the diamonds (CIS) and the triangles (CCSD(T)) are the energies of the biradical state. The CCSD(T) energies are shifted by 1.4892 au. (Reprinted with permission from Refs. [10] and [11].)...

Figure 6.10 (a) Conical intersections involved in the photochemistry of COT. MNDOC-CI energies in kcal/mol relative to the ground-state energy of COT, as well as calculated excitation energies of COT. (b) The corresponding data for the hypothetical formation of cubane from syn-tricyclo[4.2.0.02 ]octa-3,7-diene. 

TTie CIS wave function is found by solving for the coefficients c, . Since analytic gradients of CIS energies are available, one can then optimize the geometry of eadi excited state and also calculate its vibrational frequencies. CIS exdted-state geometries and vibrational frequencies are more accurate than CIS vertical excitation energies [J. F. Stanton et al.,/ Chem. Phys.,103,4160 (1995)]. 

Figure 2-10. Comparison for the depletion spectrum of Na4 and SCF-CI energies for optically allowed states and the oscillator strengths, / for a) rhombic Na4 (singlet state) and b) deformed tetradron Na4 (triplet state). The good agreement with the rhombic structnre lends confidence to the assignment of this geometrical structure. Reproduced with permission from [25]. Copyright 1991 American Chemical Sodety.

In fact, the SD-CI AEr with the 6-3IG " basis is even smaller than the HF result (a 20 kcal/mol error), while the MP2 result is only 11 kcal/mol in error (but in the other direction). However, this comparison is somewhat artificial, since few people would stop with the raw SD-CI energy difference. At minimum, the size consistency correction for the effect of quadruple excitations is usually added to the SD-CI answer. This entry is labeled est. SD(Q)-CI in Table 3 and is seen to be only 4 kcal/mol less than the full Cl limit. Other full Cl calculations have demonstrated that this simple correction works surprisingly well for small molecules. If less reliance on this correaion is desired, a whole host of MR SD-CIs can be employed that gradually approach the full Cl result. However, the size consistency error in the estimated full Cl result for CO + OH computed as a supermolecule compared to the sum of the energies of CO and OH computed separately was still 4 kcal/mol for the largest MR SD(Q)-CI wave function. 

At this point, a few remarks on the meaning of the term fully variational are in order. We take the term fully variational to imply that the conditions (16) are satisfied for all the electronic parameters that define the wavefunction. This term is introduced to avoid confusion with the term variational in the looser sense of obtained by application of the variational principle. As an example, the energy of a truncated Cl expansion is variational in the sense that the Cl coefficients have been obtained by the application of the variational principle in such a way that the calculated ground-state energy represents an upper bound to the true energy. Nevertheless, the truncated-CI energy is not fully variational since the conditions (16) hold only for variations in the Cl coefficients and not for variations in the MOs. 

The example considered above is illuminating in many ways. It is true that nobody would do a hydrogen-molecule calculation this way but the results expose some of the basic difficulties of any perturbation approach. First of all, it is sometimes necessary to work quite hard to reproduce a very trivial result. Secondly, in spite of apparent size consistency in all orders, the perturbation method does not always have the property of separability. In fact, no amount of infinite summation can give the fuIl-CI energy (i.e. the basis-set limit) once the distance between the hydrogen atoms increases beyond about twice the equilibrium bond length—simply because the expansion has a very limited radius of convergence. 

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