Physical nature of the chemical bond. IIIt A quasi-optimized 1.c.G.t.o.-m.0.-s.c.f. wavefunction for the neon hydride ion

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1 Physical nature of the chemical bond. IIIt A quasi-optimized 1.c.G.t.o.-m.0.-s.c.f. wavefunction for the neon hydride ion J. B. MOFFAT Department of Chemistry, University of Waterloo, Waterloo, Ontario Received June 19, 1968 Quasi-optimized wavefunctions for the neon hydride positive ion (NeH+) have been obtained for various sizes of basis sets of Gaussian orbitals. Theeffect of varying the size of basis set and theexponents has been examined. Canadian Journal of Chemistry, 46, 3893 (1968) Previous reports (1) of the research in quantum chemistry in this laboratory have been largely concerned with studies of nitriles. Particular attention has been given to various means for obtaining wavefunctions for nitrile molecules, both large and small. Recently the work has turned toward an analysis of the nature of the nitrile bond (2). This has been done for various reasons, some of which have been discussed previously (2). However, an additional purpose for such analysis rests on the potential applicability of such methods to the testing of the accuracy of wavefunctions. The previous paper in the present series (3) was concerned with the helium hydride positive ion. The present report on the neon hydride positive ion provides another set of wavefunctions covering a range of accuracy for the next largest molecule in the inert gas hydride ion series. The present paper is intended to provide wavefunctions for later testing as a function of the size of basis set. The molecule NeH' was chosen because of its relationship to the HeH' molecule, and for the secondary reason that very few quantum chemical calculations on NeH' have been reported in the literature. No previous calculations on NeH' with Gaussian orbitals have been found. It is well known that calculations with Slater orbitals'require smaller basis sets than those with Gaussian orbitals to achieve the same accuracy of results. Substantial efforts have been made to either circumvent or at least diminish this problem with Gaussian orbitals. Clementi (4), using a method of contracted basis sets of Gaussian orbitals, has studied a number of large molecules, including pyrrole, pyridine, and pyrazine. The alternative approach with Gaussian orbitals has been to perform calculations on large molecules using basis sets which are relatively small, at least from the standpoint of number of functions per number of electrons. Moskowitz and co-workers (5) have applied this technique to ethylene and benzene. Of course, it is assumed that these wavefunctions, built from small numbers of basis functions, will represent most of the molecular properties reasonably well. It appeared to be of interest to determine to what extent some optimization of exponents could improve these wavefunctions obtained with small basis sets. The technique of optimization used here, which is being called "quasi-optimization", involves two variables, the value of the smallest exponent in either or both of the S and P type functions, and the multiplicative factor which operated to generate the remainder of the particular set of exponents. The assumption was thus made that the exponents in any given S or P series form a geometrical progression. The advantage of this simplification was of course, that optimization by means of varying each exponent, one at a time, was avoided. In its place, the statement of the value of the initial exponent and the multiplicative factor was sufficient to define the entire set of exponents. It is not claimed that this technique is as satisfactory as the more laborious one, but rather that what may be referred to as a "quasi-optimized" function is hopefully produced in a minimum of calculations. Further it seemed reasonable to test both the effect of optimizing the smaller basis and the method of optimizing itself, on molecules small enough so that reasonably accurate results could be obtained with larger basis sets for comparison purposes. There is surprisingly little work reported in the literature on the consequences of optimizing wavefunctions of smaller basis sets of Gaussian orbitals.

2 ----P~P- ~ CANADIAN JOURNAL OF CHEMISTRY. VOL. 46, 1968 TABLE I Recent calculations on the ground 'C state of NeHi Rc -E k Method (a.u.) (a.u.) (lo5 dyne cm-') Reference Modified Platt electrostatic model United atom Two-center Hartree-Fock-Roothaan Self consistent field Gaussian orbitals Present work Finally, it was hoped that at some later time semi-empirical correlations between the best exponent values for various molecules might be obtained which would be capable of extrapolation to other molecules not originally considered. The first observation of rare-gas hydride ions was made in mass spectrometers. Because of their short lifetime it is of some interest to obtain accurate wavefunctions for these ions. The most accurate calculation on NeH' appears to be that of Peyerimhoff (6) who obtained a Hartree-Fock-Roothaan wavefunction using a basis set of 24 Slater-type orbitals (S.t.0.) centered on the nuclei. More recently, Banyard and Sutton (7) performed a united-atom (u.a.) calculation on NeH' but their results were disappointingly poor, although interesting from the standpoint of providing information on the u.a. method. The only other report of theoretical work on NeH' is that of Moran and Friedmann (8) using Platt's electrostatic model. A summary of the results of these previous workers together with the best value obtained in the present work is given in Table I. Method The Gaussian basis set calculations have been performed using the Roothaan self-consistent-field (s.c.f.) method (9) as described previously (1). A brief survey of earlier work involving Gaussian basis sets has been given (1) and need not be repeated here. The basis sets employed in the present work contained S- and P,-type Gaussian functions, of the form Nexp (-ar2) and Nq exp (-ar2), respectively, where q may be x, y, or z. The bond in NeH+ was taken to be directed along the x-axis with the neon atom at the origin. Internuclear distances between 1.45 and 2.00 bohr were employed in the present work, so as to cover the range of values for R, found by most of the previous workers, although the united atom work mentioned earlier had predicted a value of The number of S, P,, P,, and P, functions of neon are denoted by the first four figures, respectively, those of hydrogen by the second four figures, respectively, in the labelling used with the tables. The basis sets contained 2, 4, 9, 11, 17, 21, 31, 37, or 41 functions, but most of the work was done with the last three sizes mentioned. Two variables were employed in the quasi-optimization technique, the value of the smallest exponent in either or both of the S and P type functions, and the multiplicative factor which was used to generate the remainder of the particular set of exponents. Only the exponents for neon were varied. The exponents employed for hydrogen were those of Reeves (10). Results and Discussion Approximately 100 calculations were performed on the positive neon hydride molecular ion. Only the results for the largest three basis sets are reported here. Other results may be obtained from the author. All calculations have been iterated to an accuracy of hartree. Table I1 lists the values of the initial exponents and the multiplicative factors used in the present work. Table 111 shows the calculated values of electronic energy for the three largest basis sets used and for various sets of exponents. Table I11 shows that the basis set of 31 functions ( ) is the smallest set, of those basis sets employed, which yields a reasonably good energy value. The 31 function set appears to yield a minimum energy at R = 1.90 bohr, as do the 37 function and 41 function sets, in contrast with the results found earlier for HeH' using Gaussian functions (3), where the value of R, appeared to depend on the size of basis set. Table 111 further illustrates, at least for R = 1.90 bohr, that the exponents labelled NeS12, NeP4, produced the lowest energy values with the 37-function set ( ). These exponents are those which were used by Huzinaga (11) in calculations on neon with a similarly sized basis set. Exponents NeS10, NeP3 are those employed by Harrison (12) in his earlier work on neon with a basis set consisting of 9s and 5p functions. With Harrison's exponents, the present 37-function set yields an energy value approximately 0.02 hartree higher than with Huzinaga's exponents, for NeH' at R = 1.90 bohr. Approx-

3 MOFFAT: PHYSICAL NATURE OF THE CHEMICAL BOND. 111 TABLE I1 Neon atom exponents S exponents P exponents Exponent Initial Multiplying Exponent Initial Multiplying set number exponent factor set number exponent factor NeSl NePl NeS NeP NeS NeP NeS NeP NeS NeP NeS NeP NeS NeP NeS NeP NeS NeP NeSlO NePlO NeS NeS NeS NeSl TABLE 111 Effect of changes of exponents on the total energy of NeH+ at various interatomic distances and for various basis sets of Gaussian functions Size of Exponent -Total energy (hartree) at interatomic distance of basis set set numbers NeSlO, NeP NeSIO, NeP NeS 1 1. NeP imately the same energy difference exists between the results of Harrison (12) and of Huzinaga (11) for neon using the same size of basis set. The lowest value of energy for NeH' found in the present work is hartree at R = 1.90 bohr. This was obtained with the 41-function set (10, ). This is hartree above Peyerimhoff's lowest value which was found at R = 1.83 bohr. Of those sets of exponents tested with the 41-function set, Huzinaga's (11) exponents again yielded the "best" value.of course, in all the present work it is not impossible that

4 CANADIAN JOURNAL OF CHEMISTRY. VOL. 46, 1968 TABLE IV Gaussian orbital wavefunction for the ground state of NeHf (lo , lzf) at R = 1.90 bohr* Basis function CIS, C~U. K c30. K Cln.~ Ne: S S S S S S S S S S Ne: P S S H: P P P 'Obtained with a basis set of41 functions. TABLE V Orbital energies for the ground state of NeHf as obtained with various sets of Gaussian orbitals and as compared with those from the work of Peyerimhoff (6) (R = 1.83 bohr) - Orbital energies (hartree) Number of Gaussian orbitals lo n Sigma-Pi gap Pi-Pi* gapa Total energy Pi* refers to the first unoccupied Pi orbital. Number of Slater orbitals (6) 24 only relative minima have been found. However, it is believed, based on the energy values obtained, that this is not likely. Table IV shows the molecular orbital coefficients found for the Gaussian wavefunction at R = 1.90 bohr with the 41-function set. Comparison of these with those found by Huzinaga (1 1) for the neon atom shows a marked similarity, as would be expected. Table V illustrates the change in orbital energies with increasing size of basis set and compares the orbital energies obtained with Gaussian functions with those found by Peyerimhoff (6) using Slater functions. The separation between o and n orbitals together with the separation between the occupied and unoccupied n orbitals is also shown. It is readily seen that the sigma-pi gap found with any of the Gaussian sets differs substantially

5 MOFFAT: PHYSICAL NATURE OF THE CHEMICAL BOND from that found with the Slater set by Peyerimhoff. This results, of course, from the relatively large difference in n: orbital energies for Gaussian and Slater sets. Table VI illustrates variation of the distance of the center of negative charge from the "0' nucleus with R for the 41-function set. Again a substantial difference exists between the Gaussian value and the Slater value, at least for R = 1.83 bohr. TABLE VI Distance x of the center of negative charge from the Ne-nucleus for different internuclear distance R apparently possible to represent the wavefunction adequately using a 31-function basis set. Acknowled~rnents n The generous cooperation of the Computing center at the university of Waterloo and the financial assistance of the National Research Council of Canada are gratefully acknowledged. The technical assistance of Scott is greatly appreciated. See, for example, J. B. MOFFAT and R. J. COLLENS. Can. J. Chem. 45, 655 (1967), and references therein. J. B. MOFFAT and H. E. POPKIE. Int. J. Quantum Chem. 11.,~~~ 565 (1968). \ J.-B. MOFFAT. Basis set of Basis set of ~hedret. Chim. Acta, 10, 447 (1968). E. CLEMENTI. J. Chem. Phys. 46, 4725 (1967); 46, R (bohr) 41 G.t S.t.0. (6) 4731 (1967); 46, 4737 (1967). J. M. SCHULMAN and J. W. MOSKOWITZ. J. Chem. Phvs (1967): (1965): J. W. M~SKOW~TZ and M. c.' HARRISON. J. Chem. Phvs. 42, 1726 (1965); J. M. SCHULMAN, J. W. MOSKOWITZ, and C. HOLLISTER. J. Chem. Phys. 46, 2759 (1967). S. PEYERIMHOFF. J. Chem. Phys. 43, 998 (1965). K. E. BANYARD and A. SUTTON. J. Chem. Phys. 46, 2143 (1967). T. F. MORAN and L. FRIEDMAN. J. Chem. Phys. 40, In conclusion, the best Gaussian wavefunctions 860 (1964). for NeH+ obtained in this work were calculated 9. C. C: J. ROOTHAAN. Rev. Mod. Phys. 23, 69 (1951). lo. using a 41-function set and the exponents from C. M. REEVES' J. Phys S. HUZINAGA. J. Chem. Phys. 42, 1293 (1965). the earlier work of Huzinaga (11) on neon. It is 12. M. c. HARRISON. J. Chem. Phys. 41,495 (1964).

CHEM3023: Spins, Atoms and Molecules

CHEM3023: Spins, Atoms and Molecules Lecture 5 The Hartree-Fock method C.-K. Skylaris Learning outcomes Be able to use the variational principle in quantum calculations Be able to construct Fock operators