Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1963 The yh drogen molecule-ion as a superposition of atomic orbitals Allen Lowell Wasserman Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Condensed Matter Physics Commons Recommended Citation Wasserman, Allen Lowell, "The yh drogen molecule-ion as a superposition of atomic orbitals " (1963). Retrospective Theses and Dissertations. 2569. https://lib.dr.iastate.edu/rtd/2569 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. This dissertation has been 64—3905 microfilmed exactly as received WASSERMAN, Allen Lowell, 1934- THE HYDROGEN MOLECULE-ION AS A SUPER­ POSITION OF ATOMIC ORBITALS. Iowa State University of Science and Technology Ph.D., 1963 Physics, solid state University Microfilms, Inc., Ann Arbor, Michigan THE HYDROGEN MOLECULE-ION AS A SUPERPOSITION OF ATOMIC ORBITALS by Allen Lowell Wasseraan A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Physics Approved: Signature was redacted for privacy. Signature was redacted for privacy. Signature was redacted for privacy. em of Grad te College Iowa State University Of Science and Technology Ames, Iowa 1963 ii TABLE OF CONTENTS Page PREFACE iii I. HISTORICAL REVIEW 1 II. MINIMIZATION PROCEDURE 15 III. DEVIATIONS FROM THE VIRIAL RELATIONSHIP IN MANY-CENTER VARIATIONAL FUNCTIONS 25 IV. CALCULATIONS AND RESULTS 33 V. ANALYSIS OF THE COVALENT BOND IN Hg+ 51 VI. ACKNOWLEDGMENTS 96 VII. BIBLIOGRAPHY 97 APPENDIX A: EXPRESSIONS FOR MINIMIZATION PROCEDURE 105 2 2 2 APPENDIX B: RESULTS FOR THE BONDING STATES, Zg, IIu, Ag 109 2 APPENDIX C: RESULTS FOR THE ANTIBONDING STATES, Hg 170 iii PREFACE The earliest theoretical investigations of the hydrogen molecule-ion were concerned mainly with determining the validity of quantum mechanics; first the Bohr-Sommerfeld-Wilson theory and, a few years later, the Schrodinger theory of wave mechanics. Later investigations of Hg+ were motivated by the desire to obtain accurate information about experi­ mentally difficult-to-measure properties of this molecule and Hg. More recently, emphasis has shifted towards using I^+ as a model for con­ structing approximate diatomic molecular orbitals and for investigating the behavior and convergence properties of various approximations. Moreover, since is the simplest of all molecules, it has been very recently studied as part of an inquiry into the nature of chemical binding. Much remains to be done with regard to these later objectives. Complex molecular calculations are most frequently based on the LCAO (linear combination of atomic orbitals) method. Although physical and chemical intuition suggests this as a natural means of representing molecular wave functions, little is known, as yet, about the actual opti­ mal effectiveness of such superpositions. The problem is complicated by the fact that non-trivial expansions are required for three reasons: (i) The presence of several nuclei leads to molecular wave functions of peculiar geometric shapes so that, even in many-center one-electron + ++ systems, such as Hg and H3 , more than one primitive orbital per atom is needed; (ii) In many-electron systems the average shielding effect of the electron sea modifies the nuclear potential so that even the atomic self-consistent-field orbitals are superpositions of many primitive atomic iv orbitals; (iii) Correlation effects, if approached by configuration inter­ action, introduce additional atomic orbitals. Reliable knowledge about the requirements of each aspect is needed for the effective construction of electronic wave functions. The hydrogen molecule-ion represents the basic prototype for gaining insight into the geometrical aspects of LCAO expansions. An attractive feature of LCAO expansions is that they can be expected to facilitate, in a natural way, the qualitative and quantitative analysis of the difference in energy between Hg+ and (H + H*) and, thereby, contribute towards an appropriate understanding of the wave mechanical origin of chemical binding. A recent investigation has suggested that such understanding, as well as insight into complex molecu­ lar wave functions, can be gained from a suitable partitioning of the binding energy. In the case of the hydrogen molecule-ion, this par­ titioning reduces to a decomposition into promotional energy, quasi- classical energy, and interference energy; the possibilities of such an analysis can be explored without the complication of electronic inter­ actions. This is of particular interest since it was shown that, even in many-electron molecules, chemical binding essentially originates from one- electron energy contributions. Good LCAO wave functions for the hydrogen molecule-ion are, therefore, of considerable interest. However, even in this simplest of cases, only rather lengthy expressions have succeeded in accurately reproducing the molecular binding•energy. It would be desirable to have as short an expansion as possible and, in so doing, determine an order of importance of the required orbitals. The lack of such information can presumably be V attributed to the availability of direct solutions of the Schrodinger equation for the problem and to the difficulties arising in minimizing integral forms with respect to nonlinear parameters. In the present work, a number of approximations to five of the more important states of Hg+ are determined with the object of finding the shortest expansions in atomic orbitals which represent the exact eigen- functions to a considerable degree of accuracy. This is done by utilizing all parameters made available by the choice of variational functions. The basic orbitals are of the Slater type, and all superposition coefficients, as well as orbital exponents, are independently varied in order to obtain the lowest possible energy expectation value. The nonlinear minimizations are carried out by a somewhat novel, convergent, iterative technique developed for this purpose. Thus, accurate representations are obtained 2 2 2 2 ant 3 for the lowest £g> ^u» ^u» ^g' * Ag states. A new stable state of the molecule is found: The energy curve of the lowest 3Ag state is shown to have a minimum at about 9.8 A with a binding energy of about 0.36 kcal. The wave functions obtained are analyzed on the basis of the binding energy partitioning mentioned earlier. The examination of the energy components, particularly as functions of the intemuclear distance, reveals considerable regularity and similarity among the various states. Thus, a better understanding of the wave functions, as well as the origin of the one-electron chemical bond, is achieved. Throughout this work, except when otherwise noted, energy and length are reported in atomic units (38, 102): 1 Hartree = 1 H = (me4/h3) = 27.210 ev vi 1 Bohr = 1 b = (h3/me3) = 0.529171 x 10"8 cm, where m is the mass of an electron and e is its charge. In order to compare theory with experiment, the following conversion factors are needed (38): Hydrogen Rydberg = R^ = 109 677.576 cm-1 Infinite mass Rydberg = R^ = 109 737.309 cm-1 Experimental ionization energy = 1^ = 13.5978 ev of the hydrogen aVom Nonrelativistic, infinite- = I *= 13.605 ev = 0.5 H nuclear-mass approximation to the hydrogen atom ionization energy Three different sets of nomenclature are in use for identifying the states of Hg+ . They correspond as follows: Spectroscopic United atom Separated atom notation symbol symbol lowest 32 CT lS g 1Sag g lowest 3S ls u 2^u au 3 lowest n 2Prr nu2p u u lowest 2n p g 3dng V 3 3d6 6 3d lowest Ag g g second 3Z 2sct a 2s g g g 2 third Eg 3dag ag3s second 3II u etc. In the case of Hg+, the spectroscopic notation presents some ambiguities because of violations of the noncrossing rule between states belonging to the same irreducible representation. Accurate calculations vii show that the 2sa and 3da states, and also the 3pn and 4£tt states g g u u cross (4, 108). There is, however, no ambiguity with respect to the lowest level of each symmetry, with which the present calculations are concerned. 1 I. HISTORICAL REVIEW A. Experimental Investigations Obtaining experimental information about the hydrogen molecule-ion is complicated by the fact that no discrete spectrum of this positive ion has ever been observed. [Brasefield (17), in 1927, observed 45 weak emission lines which he identified as Hg+ rotational bands, but after examining the energy curves calculated by Morse and Stuckelberg (83), he recanted (16). He did, however, maintain that some strong continuous spectra were due to the hydrogen molecule-ion.] That I^+ did exist as a stable molecule was discovered as early as 1907 by J. J. Thompson in canal>rays (112). In a discharge tube filled with hydrogen, he observed positive ions with a mass to charge ratio of 2, corresponding to I^+. Similar observations were reported by Knipp (70) and by Dempster (33). With improved canal ray techniques it became possible to measure the appearance potential of Hg+ or, equivalently, the ionization potential of the hydrogen molecule. This value was variously reported as 11.0, 13.5, 16.0, 16.68, and 22.8 volts, with 16.0 appearing to be most popular. The value 22.8 volts, claimed by Horton and Davies (58) in 1923, was regarded by them a confirmation of Bohr-Sommerfeld- Wilson type quantum theoretical calculations made by Pauli (86) and Niessen (8 5) a year earlier. But in the same year, Smyth (104, 105), using a hydrogen filled discharge tube at the very low pressures which favored Hg+ formation to the exclusion of other products, found its appearance potential to be 15.9 volts (.5843 H), in agreement with the earlier consensus.