A local Schrödinger equation (LSE) method is proposed for solving the Schrödinger equation (SE) of general atoms and molecules without doing analytic integrations over the complement functions of the free ICI (iterative-complement-interaction) wave functions. Since the free ICI wave function is potentially exact, we can assume a flatness of its local energy. The variational principle is not applicable because the analytic integrations over the free ICI complement functions are very difficult for general atoms and molecules. The LSE method is applied to several 2 to 5 electron atoms and molecules, giving an accuracy of 10 ÿ5 Hartree in total energy. The potential energy curves of H 2 and LiH molecules are calculated precisely with the free ICI LSE method. The results show the high potentiality of the free ICI LSE method for developing accurate predictive quantum chemistry with the solutions of the SE.
Inverted sandwich type chromium(I) complexes, (mu-eta(6):eta(6)-C(6)H(5)CH(3))[Cr(DDP)](2) (DDPH = 2-(4-{(2,6-diisopropylphenyl)imino}pent-2-ene) and (mu-eta(6):eta(6)-C(6)H(5)CH(3))[V(DDP)](2), synthesized by Tsai et al., and (mu-eta(6):eta(6)-C(6)H(6))[Cr(DDP)](2) synthesized by Monillas et al., were theoretically investigated with MRMP2 and DFT methods, where model compounds (mu-eta(6):eta(6)-C(6)H(6))[Cr(AIP)](2) (AIPH = (Z)-1-amino-3-imino-prop-1-ene) were mainly employed. Both computational methods succeeded in reproducing the experimental facts that the chromium and vanadium complexes take surprisingly high spin states, septet and quintet spin states, respectively. MO diagrams of these complexes present a clear understanding of the reasons why they take such high spin states. We also calculated their analogues, (mu-eta(6):eta(6)-C(6)H(6))[M(DDP)](2) (M = Sc, Ti, Mn, or Fe). The spin multiplicities of the Sc and Ti complexes were calculated to be singlet and triplet, respectively, by the DFT(B3LYP) method. Those of Mn and Fe complexes were calculated to be quintet and triplet, respectively, by the DFT(B3LYP) method, but nonet and singlet, respectively, by the MRMP2 method, suggesting that the DFT method cannot be applied to these complexes. The MRMP2 calculations indicate that the spin multiplicity increases upon going to Mn from Sc and reaches the maximum, nonet spin state, at Mn, and then suddenly decreases to singlet at Fe. This interesting change in spin multiplicity is discussed in terms of occupation of MOs.
The free iterative-complement-interaction ͑ICI͒ method based on the scaled Schrödinger equation proposed previously has been applied to the calculations of very accurate wave functions of the hydrogen molecule in an analytical expansion form. All the variables were determined with the variational principle by calculating the necessary integrals analytically. The initial wave function and the scaling function were changes to see the effects on the convergence speed of the ICI calculations. The free ICI wave functions that were generated automatically were different from the existing wave functions, and this difference was shown to be physically important. The best wave function reported in this paper seems to be the best worldwide in the literature from the variational point of view. The quality of the wave function was examined by calculating the nuclear and electron cusps.
We introduce here the exponential integral (Ei) function for variationally solving the Schrödinger equation of helium and its isoelectronic ions with the free iterative complement interaction (ICI) method. In our previous study [J. Chem. Phys., 2007, 127, 224104], we could calculate very accurate energies of these atoms by using the logarithmic function as the starting function of the free ICI calculation. The Ei function has a weak singularity at the origin, similarly to the logarithmic function, which is important for accurately describing the three-particle coalescence region. The logarithmic function, however, has a node and a maximum along the radial coordinate which may be physically meaningless. In contrast, the Ei function does not have such unphysical behaviors and so would provide an improvement over the logarithmic function. Actually, using the Ei function, instead of the logarithmic function, we obtained the energy, E= -2.903 724 377 034 119 598 311 159 245 194 404 446 696 924 865 a.u. for the helium ground state with 21 035 functions, which is a slight improvement over our previous result (the bold face shows the digits that are believed to have converged). This result supports the suggestion that the Ei function is better than the logarithmic function for describing the three-particle coalescence region.
We study the electronic structure of two types of transition metal complexes, the inverted-sandwich-type and open-lantern-type, by the electronic stress tensor. In particular, the bond order b ε measured by the energy density which is defined from the electronic stress tensor is studied and compared with the conventional MO based bond order. We also examine the patterns found in the largest eigenvalue of the stress tensor and corresponding eigenvector field, the "spindle structure" and "pseudo-spindle structure". As for the inverted-sandwich-type complex, our bond order b ε calculation shows that relative strength of the metal-benzene bond among V, Cr and Mn complexes is V > Cr > Mn which is consistent with the MO based bond order. As for the open-lantern-type complex, we find that our energy density based bond order can properly describe the relative strength of Cr-Cr and Mo-Mo bonds by the surface integration of the energy density over the "Lagrange surface" which can take into account the spatial extent of the orbitals. Wave function analysis; Theory of chemical bond; Stress tensor; Transition metal complexes arXiv:1109.3063v1 [physics.chem-ph] 14 Sep 2011The stress tensors in quantum systems have been investigated for many years, including one of the earliest quantum mechanics papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The stress tensors in general are widely used for description of internal forces of matter in various fields of science such as mechanical engineering and material science. As for the stress tensors in quantum mechanics context, we can find several different definitions and applications in the literature. For example, Ref.[6] and followers focus on the stress tensor which is associated with forces on nuclei. In contrast, the one we consider in this paper is the electronic stress tensor, which is associated with effects caused by internal forces acting on electrons in molecules, following Ref. [12]. This electronic stress tensor has been used to investigate chemical bonds and reactions and many interesting properties have been discovered [12,15,16,[20][21][22][23][24][25][26][27].Among them, it is shown that the energy density can be defined from the electronic stress tensor. Using this energy density, new definition of bond order is proposed [21]. So far, this stress-tensor-based bond order is applied to s-block and p-block compounds in Refs. [21,22,27] and found to have reasonable features. Then, next question is whether this bond order would work well for d-block compounds.In this paper, we wish to address this issue using two types of transition metal complexes. The first one is the inverted-sandwich-type dinuclear transition metal complexes and the second one is the open-lantern-type dinuclear transition metal complexes. Based on the electronic structures that are thoroughly investigated in Refs.[28] and [29], we study the electronic stress tensor of these molecules. Our special attention is given to chemical bonds between metal atoms and benzene for the...
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