Robert S. Mulliken

Electronic Population Analysis on LCAO–MO Molecular Wave Functions. II. Overlap Populations, Bond Orders, and Covalent Bond Energies

LCAO molecular orbital overlap populations give in general much more flexible and widely useful measures of the non‐Coulombic parts of covalent bond energies than do LCAO bond orders. They are immediately applicable to both π and σ bonds, including bonds involving hybrid AOs of all kinds, and they take account directly of the effects of variations in bond length on bond strength. In the last section of this paper, a number of ways of defining LCAO bond orders are reviewed, and their advantages and disadvantages discussed.If all LCAO parameters β are assumed proportional to corresponding overlap integrals S times suitable mean atomic ionization energies Ī, a simple general approximate formula for covalent resonance energies is obtained in terms of partial overlap populations and Īs, including one or two empirical coefficients. This formula indicates that forced hybridization (see III of this series) due to inner shells should make important negative contributions to bond energies. The application of the f...

Formulas and Numerical Tables for Overlap Integrals

Explicit formulas and numerical tables for the overlap integral S between AOs (atomic orbitals) of two overlapping atoms a and b are given. These cover all the most important combinations of AO pairs involving ns, npσ, and npπ AOs. They are based on approximate AOs of the Slater type, each containing two parameters μ [equal to Z/(n—δ)], and n—δ, where n—δ is an effective principal quantum number. The S formulas are given as functions of two parameters p and t, where p=½(μa+μb)R/aH , R being the interatomic distance, and t=(μa—μb)/(μa+μb). Master tables of computed values of S are given over wide ranges of p and t values corresponding to actual molecules, and also including the case p=0 (intra‐atomic overlap integrals). In addition, tables of computed S values are given for several cases involving 2‐quantum s, p hybrid AOs.Hybrid S values for any desired type of hybrid can be obtained very easily from the tables as simple linear combinations of non‐hybrid S values. It is shown how S values correspondin...

Report on Notation for the Spectra of Polyatomic Molecules

A SATISFACTORY standardization of notation . for diatomic molecules and their spectra already eXIsts. l The problem of standardization for polyatomic molecules is much more complicated, because of the greater number of degrees of freedom for rotation and vibration, and because of the considerable variety of types of. symmetry, or symmetry groups, to which polyatomlC molecules belong, as contrasted with only two types (Coov and Dooh) for diatomic molecules. Fortunately, the system used in Herzbergs well-known book on infrared and Raman spectra2 is already very generally accepted, and forms a suitable basis for the standard notation proposed here. However, when one considers the entire range of polyatomic spectroscopy from ultraviolet to microwave and magnetic resonanc~ spectra, there are several points of notation which need clarification, revision, supplementation, or simply codification and affirmation in a form to which spectroscopists of all wavelengths can conveniently refer. The principal need for supplementation, as compared with Herzbergs book, arises from the consideration of molecules in nontotally-symmetric electronic states, a matter which is especially essential for electronic spectra. In some points the problem of notation for polyatomic spectra overlaps that for other related areas; in particular, valence theory, and the states of atoms in crystals. Some consideration has been given in this report to the needs of these areas, and a few recommendations have been made. The problems are rather different for small molecules the high symmetry of which is often very important fo; the classification of vibration modes and quantized states, and large molecules, where rigorous symmetry is often nonexistent or unimportant (though approximate or local symmetry is often important). Some consideration has been given in this report to large molecules, but the major emphasis is on the smaller or more symmetrical molecules .

Electronic Population Analysis on LCAO‐MO Molecular Wave Functions. IV. Bonding and Antibonding in LCAO and Valence‐Bond Theories

It is shown that there is a practically one‐to‐one correspondence between the occurrence, on the one hand, of positive (bonding) and negative (antibonding) overlap populations in LCAO theory and, on the other hand, of bonded attractions and nonbonded repulsions in VB (valence‐bond) theory. This correspondence is discussed in terms of examples, and is traced for the N2 molecule both for the assumed case of no s–p hybridization, and for the actual case with hybridization. It is pointed out that repulsions between nonbonded atoms in VB theory (including those which give rise to steric hindrance) have their counterpart in negative overlap populations between the same atoms in LCAO theory. The π overlap populations for the various links in 1,3‐butadiene are computed by LCAO theory. It is shown how they are affected by conjugation (see Table I) and the results are compared with those of VB theory.

Criteria for the Construction of Good Self‐Consistent‐Field Molecular Orbital Wave Functions, and the Significance of LCAO‐MO Population Analysis

Criteria for the optimal choice of finite linear combinations of STOs (Slater‐type orbitals) adequate to closely approximate SCF (self‐consistent‐field) MOs are examined in the light of computations on HF and other molecules. First, however, some aspects of the AO (generalized Heitler—London) method are discussed. The concept of induced configuration mixing is proposed as a helpful way of thinking of the higher‐order interactions which carry the wave function from that of a ground‐state atom pair over toward an SCF‐MO form. The concept of valence polarization is suggested as preferable to and/or more general than that of hybridization. When, in the AO method, a molecule is formed from atoms, the AOs of the latter are valence polarized, and if the molecule is heteropolar, also Coulomb polarized. It is noted that both types of polarization are effected in part by the mixing in of highly shrunken excited AOs.The same concepts are applicable also in the MO method. In an optimal choice of a finite STO basi...

Potential Curves of Diatomic Rare‐Gas Molecules and Their Ions, with Particular Reference to Xe2

The excited and ionized states of the heavier rare‐gas molecules are discussed, existing evidence on the dissociation energies of the ions are reviewed, and estimates of these dissociation energies are made. Estimated potential curves for the Xe2+ ion and for the lower excited states of the Xe2 molecule are given. Applications to the interpretation of the observed spectra of the heavier rare gases, especially Xe2, are discussed.

Intensities in Molecular Electronic Spectra X. Calculations on Mixed‐Halogen, Hydrogen Halide, Alkyl Halide, and Hydroxyl Spectra

The dipole strengths for certain perpendicular‐type transitions N→Q in the mixed halogens, the hydrogen and monovalent metal halides, and the alkyl halides, are calculated theoretically by the LCAO and by the AO approximations. The scanty experimental absorption coefficient data on these halides (particularly the hydrogen and alkyl bromides and iodides) are critically examined, and acceptable experimental dipole strengths are obtained for the bromides. These show, very gratifyingly, the same kind of agreement with the calculated values as was found in IX of this series for the N→Q transitions in F2, Cl2, and Br2. The iodides, however, just like I2, show anomalously low strengths for the true N→Q part of the intensity (N→3II1 and N→1II), together with high strength for N→3II0+. These anomalies are ascribed here, as in I2, to partial case c coupling (partial preservation of atomic Js). The comparison between theory and experiment confirms the interpretation of the ultraviolet continua of the hydrogen and a...

Intensities of Electronic Transitions in Molecular Spectra I. Introduction

The neglected problem of the theory of the absolute intensities of electronic transitions in molecular spectra is discussed. General equations for dipole strength, Einstein coefficients, mean lifetimes of excited states, f values, and so forth, are collected in forms convenient for use in later papers where the theory for various types of transitions will be developed and applied.

Far Ultraviolet Absorption Spectra of Ethylene and Ethylene‐d4

The absorption spectra of ethylene and ethylene‐d4 have been reinvestigated in the 1500 A to 2050 A region, using the first order of the Harrison 21‐foot vacuum spectrograph. Vibrational constants of the C=C stretching vibration in the R (first Rydberg) state were determined to be: C2H4, ω20=1381.5 cm—1; x220=—11.2 cm—1; C2D4, ω20=1306.8 cm—1; x220 —8.2 cm—1; and (tentatively) in the V state: C2H4, ω20=852 cm—1; x220=—1.9 cm—1; C2D4, ω20=797 cm—1; x220=—1.8 cm—1. Newly observed fine structure in the v′—0 stretching vibration progression in the long wavelength part of the V←N transition of C2D4 is tentatively attributed to a torsional oscillation. The 0—0 bands of the V←N transition, though too weak to be seen, were estimated to lie near 2500 A. The probable dissociation of ethylene into CH2 radicals in the V←N continuum is discussed.The torsional frequency in the R state was redetermined as 141 cm—1 (C2D4) and 236 cm—1 (C2H4), decreasing somewhat with increasing stretching quantum number. An attempt has b...

Intensities of Electronic Transitions in Molecular Spectra II. Charge‐Transfer Spectra

Two related types of high intensity molecular spectra are discussed. These are designated as charge‐resonance spectra and charge‐transfer spectra. In the molecular orbital approximation, both these types usually involve transfer of one electron from a bonding to the corresponding antibonding molecular orbital. In the atomic orbital approximation, charge‐resonance spectra involve a transition from a symmetrical to a corresponding antisymmetrical state, the wave function of each of these two states being a linear combination of two wave functions representing two hypothetical states of a molecule that are equivalent or nearly so but different in charge distribution. Charge‐transfer spectra in the atomic orbital approximation involve a transition from a state having a non‐ionic to one having an ionic wave function (N→V transition). Theoretical equations for dipole strengths and f values for both these types are given, in each case calculated according to each of the two approximations. The equations are very...