On the Structure of Mo(CN)8−4

Racah, Giulio

doi:10.1063/1.1723829

Cited by 28 publications

(9 citation statements)

References 5 publications

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“…The absolute maximum of SSPd = 2.9884 occurs at a1 = 76.32° (Fig. 7)-that is, when the corners of the tetragonal antiprism are at an angle of 60.90°with the tetragonal axis, in agreement with Racah's result for the best set of eight equivalent spd hybrid orbitals (8).…”

Section: Tetragonal Pyramid (C4v)supporting

confidence: 78%

Reliability of the pair-defect-sum approximation for the strength of valence-bond orbitals

Pauling¹,

Herman²,

Kamb³

1982

Proc. Natl. Acad. Sci. U.S.A.

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The pair-defect-sum approximation to the bond strength of a hybrid orbital (angular wave functions only) is compared to the rigorous value as a function of bond angle for seven types of bonding situations, with between three and eight bond directions equivalent by geometrical symmetry operations and with only one independent bond angle. The approximation is seen to be an excellent one in all cases, and the results provide a rationale for the application of this approximation to a variety of problems.The quantum mechanical treatment of the electronic structure of molecules has great value. For all except the simplest molecules, it is so difficult to carry out a thorough calculation as essentially to prevent its application. One simplification in the theoretical discussion of the nature of the bonds formed by an atom is to assume that the bond-forming strength of an atomic orbital is proportional to the amount of overlap of the orbital with a bond orbital of an adjacent atom. The amount of overlap increases with increase in the concentration of the bond orbital in the bond direction. A second approximation was made by one ofus in 1931. This approximation is based on the knowledge that the radial parts of the atomic orbitals involved in the formation of hybrid bond orbitals are not much different for the different atomic orbitals, so that the assumption that they are the same does not lead to great error. Accordingly attention is, focused on the angular distribution of the bond orbitals. The strength, S, of a bond orbital was defined as the value in the bond direction of the angular part of the bond orbital, normalized to 4ir over the surface of a sphere (1, 2). A number of sets of bond orbitals were discussed in this way (1, 2). Hultgren then attacked the general problem of the best possible sets of sp3d5 orbitals (3). The mathematical problem was, however, too difficult to handle before computers had been developed, and Hultgren made a simplifying assumption, the assumption of cylindrical symmetry of the bond orbitals. This assumption led to the erroneous conclusion that sets of more than six bond orbitals would not be well suited to bond formation. Nearly 40 years later the general problem of the best set of nine spd bond orbitals was attacked by McClure (4). He found one excellent set of nine hybrid bond orbitals, all with strength close to the maximum possible, 3. Because of the difficulty in carrying out the calculations rigorously for large sets oforbitals, an approximation was then suggested. This approximation is based on the equations giving the strength SO of two equivalent bond orbitals that have the maximum strength in directions making the angle a with one another. For sp3, sp3d5, and sp3d5f7 hybrid orbitals the equations for the bond strength SO (a) have been derived (5, 6)-Eqs. 1, 2, and 3, respectively. [2] [3] where x = cos2 (a/2). For sp3 hybrid orbitals, the maximum strength Sma=, 2, is found at the tetrahedral bond angle 109.470; for sp3d5 hybrid orbitals, Smax = 3 at a = 73.15°and a...

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Section: Tetragonal Pyramid (C4v)supporting

confidence: 78%

Reliability of the pair-defect-sum approximation for the strength of valence-bond orbitals

Pauling¹,

Herman²,

Kamb³

1982

Proc. Natl. Acad. Sci. U.S.A.

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“…These two arrangements are shown in Figure 12. For the tetragonal antiprism, the maximum in the bond strength S occurs when the corners are at an angle of 60.90" with the tetragonal axis, as found by Racah [6], compared to an angle of 59.26" for the Archimedian antiprism. At this angle of 60.90', the exact bond strength is 2.9884 (Table I), while application of Eq.…”

Section: Octacovalencementioning

confidence: 63%

“…This group theoretical approach was extended to spdf hybrid orbitals by Shirmazan and Dyatkina [4] in 1953 and by Eisenstein [5] in 1956. Meanwhile, the theory of hybrid bond orbitals was applied to elucidate the structure of Mo(CN):by Racah [6] in 1943, to quadricovalent complexes of the transition metal ions Ni 11, Pd 1 1 , Pt 11, Au 1 1 1 , Ag 11, and Cu I I by Kuhn [7] in 1948, to bipyramidal heptacovalent complexes, such as IF,, by Duffey [8] in 1950, to the origin of the potential barriers to internal rotation in molecules by Pauling [9] in 1958, and to bonding in icosohedral complexes by Macek and Duffey [ 101 and in cuboctahedral complexes, such as UBI2, by Canon and Duffey [ 1 11 in 1961. A thorough discussion of the applications of the theory of hybrid bond orbitals and valence-bond theory is presented in the third edition of The Nature of the Chemical Bond [ 121, published in 1960. During the last score of years there has been intense experimental activity in the fields of synthesis and characterization of compounds of the transition metals and of the lanthanide and actinide elements.…”

Section: Introductionmentioning

confidence: 99%

Recent advances in simple valence‐bond theory and the theory of hybrid bond orbitals

Herman¹

1983

Int J of Quantum Chemistry

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The history of important developments in the theory of hybrid bond orbitals and its application in valence-bond theory is reviewed. One of the salient points, the bond strength of a hybrid orbital. is defined as the value of the angular part of the orbital along the bond direction. Characteristic bond angles corrcsponding to maxima in the bond strength are presented for various basis sets. In order to alleviatc computational difficulties in determining the bond strength for complex systems, an approximation to it. the pair-defect-sum approximation, is described. The results of an exhaustivc tcst of thc validity of this approximation are presented. AppticdJions of these ideas to coordinate chemistry are provided. Finally. the case is made for the continued viability of valence-bond theory in this age of omnipresent computer terminals.

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“…Angles between planes 1 and 2 87.7 °, 1 and 3 1"2 °, 2 and 4 2.3 °. (1)-Mo(1)-As(1)-C(18) 56.5 3-28 CI(1)-Mo(1)-As(1)-C(19) -66.3 3.51 Cl(2)-Mo(1)-As(2)-C(28) -62.0 3.37 Cl(2)-Mo(1)-As (2) Racah (1943).…”

Section: Discussionmentioning

confidence: 99%

Crystal and molecular structure of tetrachlorobis[ophenylenebis(dimethylarsino)]molybdenum(V) triiodide

Drew¹,

Egginton²,

Wilkins³

1974

Acta Crystallogr Sect B

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On the Structure of Mo(CN)8−4

Cited by 28 publications

References 5 publications

Reliability of the pair-defect-sum approximation for the strength of valence-bond orbitals

Reliability of the pair-defect-sum approximation for the strength of valence-bond orbitals

Recent advances in simple valence‐bond theory and the theory of hybrid bond orbitals

Crystal and molecular structure of tetrachlorobis[ophenylenebis(dimethylarsino)]molybdenum(V) triiodide

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