University of Chicago Laboratory of Molecular Structure and Spectra Technical Report: 1968: Searching Inside

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... scaling) Liu, Lyu,,, and Byers Brown13 determined the energy through second order at the equilibrium... separation which corresponded to an energy of dissociation equal to 4.988 eV= 40233 cm-1 or 105...% of the experimental value; whereas this same basis set gives a Rayleigh-Ritz variational energy corresponding to 95...% of the experimental value. Goodisman'4 used a 10-term function to calculate the energy through the third order... to obtain a dissociation energy equal to 4.617 eV= 37 241 cm-' or 97.2% of the experimental. Matcha

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... EXCHANGE AND COULOMB ENERGY OF H, one or else energy optimized) are calculated using four... are comparatively good, but at large separations the EL-HAV second-order energy becomes approximately one-half... of the accurate dispersion energy. The energy of the triplet state, added to or subtracted from the energy... of the singlet state, yields the Coulomb or exchange energy, respectively. A surprise to us... is that the perturbation procedures give better values of the exchange energy than of the Coulomb energy. To test whether

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... and Byers Brown [38] the energy of H2 was calculated using a high-order Rayleigh-Schrodinger perturbation... as a perturbation. At small values of R (0.4 < R < 0.8) the energy was calculated up to the eighth order... over the variational method in energy calculations. The complexity of the numerical calculations... always an upper bound to the true energy. Various formalisms proposed recently [12, 40-43... the interaction energy of two ground-state hydrogen atoms. Both the XL+ and the b3E, states were considered

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... for this situation is roughly that the energy differences between planar and perpendicular ethylene depend only... on that part of the energy of formation of C2H4 which is provided by the interaction of the two 2px electrons... of the two CH2 groups. In perpendicular C2H this energy is small (hyperconjugation energy only), and the V B2... in the configuration (2e)2 In planar C2H4 in its ground state, *..(x + x)2, this energy is that of the w bond, which... gives a stabilization energy of 2.8 eV. In excited planar C2H4, ** (x+x)(x-x), we expect the antibonding

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.... A. Exchange and Coulomb Energy of H2 Determined by Various Perturbation Methods* P. R. CERTAINt... of the hydrogen molecule at internuclear separations R= 4,6, 8ao. The energy through second order... are good at small separations but at large separations they give a second-order energy which appears... to be about one-half the correct dispersion energy. The Rayleigh- Schrodinger treatment using a Sternheimer... type of zeroth-order Hamiltonian gave the best energy for the ground state but not very good energy

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... into the total wavefunction to give the most significant energy improvement. In the case of the two-electron... additional energy lowering. For example, the energy improvement in He due to inclusion of the 2s.... Again, once the configuration 2p is included, the addition of the 39 configuration improves the energy... correlation energy is obtained, then the addition of further configurations improves the energy only little..., at a point where any additional computational labor gives diminishing returns with respect to energy

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... CERTAIN, HIRSCHFELDER, KOLOS, AND WOLNIEWICZ second-order polarization (or dispersion) energy E(2...) approaches E(2) (dis), however. We point out that we have evaluated the second-order energy with a truncated... wavefunctions. Thus we can only say that it appears that the Eisenschitz-London or HAV second-order energy... is asymptotically equal to one-half of the second-order dispersion energy. B. Arguments that iE(2) (EL-HAV)~IE(2... perturbation schemes to the model leads to an expansion for the energy consisting of a power series in R-1 plus

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... of the dominant configuration are most important., ) The error trend, i.e., the part of the correlation energy... SYSTEMS Recently there have been several calculations on the correlation energy of the four-electron... a configurationinteraction point of view, the separation of the total correlation energy into intra- and intershell...' with ls2(2p)2a2 yields E2s= -0.04370 a.u. as correlation energy for the L shell. Addition of the configuration... 2s2(3p) ,.' results in an energy lowering of el.= -0.01864 a.u., which should be attributed

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... are the "predicted" values. In these expressions, a is the actual energy separation which arises from experimental... results (for AH this energy value is the separation between the respective minima of the two potential..., the differences are taken so that a > 0. The corresponding energy separation calculated to Hartree-Fock... approximation is designated 6EHF, 6Ecorr is the change in the correlation energy between the two states... of the UA or AH, and Er,,, is the change in the relativistic energy between the two states of the UA or AH

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...) to learn the nature of molecular phenomena (character of interaction, crossing of potential energy curves... been performed [1, 2] and they show that for the lowest vibrational state the nonadiabatic energy... quanta may be due to the nonadiabatic effects. However, the nonadiabatic energy correction.... On the other hand, for muonic molecules m/Mt~ 0.1, and the nonadiabatic energy correction becomes very... important [3-5] amounting to 1 % and 0.1 % of the total energy for p4p and taut, respectively

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... EXCHANGE AND COULOMB ENERGY OF H, tials. The Sternheimer potential energy function is Uo,= e- (KA... with Eq. (13). Again making use of Eq. (20), the Rayleigh-Schridinger second-order energy is 'E(2)(RS) =... the Hylleraas variational principle'2,8 to determine an upper bound for the second-order energy 'E(2) (RS...) is equal to the secondorder energy of Eq. (23) if the exact ,t1) (RS) is replaced by o(')(RS). In our..., X) )/((1, X) I19(1, X)), (26b) should then give the energy accurate through the third order

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... EXCHANGE AND COULOMB ENERGY OF H2 TABLE VIII. Comparison of perturbation exchange energies... the exchange energy through the second the first order is Eezxeh(1, X). The values given are half... of the difference order of perturbation Is E..hs; and the exchange energy obtained from the expectation values... of 'E(2) calculated by the HS, MS-MA, and RS are approximately equal to the dispersion energy E(')(dis... the average excitation energy must be greater than 0.75, the energy required to excite the two hydrogen atoms

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... energies are corrected for the "change in correlation energy" involved in the transition making use... of united-atom changes in the correlation energy. The singlet-triplet (a 1A-X 3E-) intercombination separations... on a quantitative basis. Thus, from experimental results for the appropriate UA energy separation ('D-3S...), this approach is used to predict energy separations between the X 2E+, A 2I, (B)2Ar energy levels..., and is extended to include properties other than energy separations. These studies have indicated that rather

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... them were limited to the inner minimum of the potential energy curve. A complete calculation has..., they still fail to account for a sizable portion of the correlation energy, especially in the vicinity... in the region of the avoided crossing of the potential energy curves. II. POTENTIAL ENERGY CURVE 13,14

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... region of the internuclear distance. Since at large R a second avoided crossing of the potential energy..., decreases with increasing internuclear distance. For R ) 6 the potential energy curve for the excited 1>+ g.... However, the B state dissociates into is and 2p hydrogen atoms, and the interaction energy decreases like... potential energy curves increases with increasing R. 67

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... approximation, and the solid ones those resulting from the adiabatic potential energy curve. The error... potential energy curve employed in the calculation. However, an inaccurate potential energy should cause... roughly the same error in all vibrational levels located in the same energy region, independently

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... within the first order approximation. The exciton splitting energy due to exchange of excitation energy... appears in the interaction of the G state with the CT state. Energy separations between the observed CT... ground state by interaction with the CTg state. The exciton splitting energy has to be considered, which... is reduced to the transition dipoledipole approximation. The splitting energy is shown as a function... in fig. 1. The splitting energy and transition moments of the exciton band in the dimer can be expressed

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...- and 63-term wavefunctions were almost identical. Thus at R = 3.0 the energy obtained with the 63-term... of terms in this region would certainly result in a somewhat lower energy and would decrease the height... of terms in the expansion, D the bind- -1 ing energy in cm , E and V the total and potential energies... in a.u., and dE/dR the derivative of the total energy with respect to the internuclear distance obtained

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... of triphenylene. Thus an energy less than 66 kcal/mole for the 900 twisted olefine may still... of styrene as a photosensitizer with butene-2 may not exclude an olefine triplet energy lower than 62... in conversation, the energy is so high here that the olefine could still be nearly planar during the energy

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... Rydberg MO's in order of energy are expected to be 3blu (3p6 in UAO) and 2b2u(3p7r ), but these should... states lies at 7.45 eV have been found by Ross and Lassettre76 in the energy loss spectrum of ethylene... a detector after exciting a gas molecule and losing the corresponding excitation energy. When the electrons... between these energy loss spectra and the frequencies absorbed in ordinary ultraviolet 169