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Quantum-Mechanical Treatment of Helium-Hydride Molecule Ion Heh+

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Quantum-Mechanical Treatment of Helium-Hydride Molecule Ion Heh+ Quantum-Mechanical Treatment of Helium-Hydride Molecule Ion HeH+. By Soroku TOH. (Read Dec. 16, 1939by M. KOTANI). § 1. Helium-hydride molecule ion HeH+, whose existence and process of formation was investigated experimentally by mass-spectro- graphic analysis(1), is the simplest of the stable heteronuclear diatomic molecules. The structure of this molecule-ion was investigated theore- tically by Beach`', using modified atomic orbitals, and he obtained 0.5 e.V. as the lower limit of dissociation energy . While our calculation was in progress, Coulson and Duncanson published an interesting paper"', in which the various approximate wave-functions for HeH+ were critically studied in detail. They obtained 0.8 e.V. (0.0297 a.u.)(4) as the lower limit of dissociation energy and estimated its probable value as 1.0~1.5 e.V.. lit this paper we applied to this ion the method of James and Coolidge, which gives in case of H2 molecule very good results(5). Coulson and Duncanson have already applied this method, but since the number of terms was rather small they obtained no more than 0.32 e.V. as the dissociation energy in this method. Because the knowledge about the structure of HeH+ is almost entirely afforded from the theoretical side, it is desirable to extend the investigation, and to derive various constants of this ion theoretically as closely as pos- sible. This is attempted in the present paper. §2. The Hamiltonian operator of our problem is, in atomic units, where ra, rb are distances of electron from He and H nuclei resptectivly. Our task is to minimize H d 1d 2/ d 1d 2, by choosing suitable approximate functions for T In James-Coolidge's method (1) M'Ewen and Arnot, Proc. Roy. Soc. A 172 (1939), 107. (2) J. Y. Beach, J. Chem. Phys 4 (1936), 353. (3) Coulson and Duncanson, Proc, Roy. Soc. A 165 (1938), 90. (4) Energy value without allowance for zero point enemy is cited here for the sake of comparison with our result. (5) James and Coolidge, J. Chem. Phys. 1 (1933), 825. 120 Soroku TOH. [Vol. 22 is taken as a linear combination of functions of the type where =2/br12, and , are elliptic coordinates: The symmetry of the function (1) corresponds to 'Y, just as in case of the ground state of Hz. However, since we have no symmetry plane perpendicular to the internuelear line in case of HeH+, j+1.- may take any integral values, so that, the number of important terns to be taken in is fairly large. As a preliminary study we have taken the wave function: This i is almost the simplest molecular orbital for HeH+. Taking c=1.250, 1.375, 1.500 and 1.625, we determined k so as to minimize for certain range of R. The result H d 1dis 2/as d 1d 2 shown in Fig. 1. Now the equilibrium nuclear distances obtained by Beach and Coulcon-Duncanson are 1.57 and 1.446 respectively. Fig. 1 shows that c=1.375 gives the lowest clergy in the neighbourhood of =1.50. Taking this into account we have chosen the parameter c inR (1) as 1.375. §3- For the choice of r, we have another method of estimation as follows: Since tho energy of He+H is higher than that of He+H+, HeH+ dissociates into urinal He atom and proton as R tends to . In this limiting case the approximate wave function is given by with =37/16. On the other haul, HeH+ be- 1940] Quantum-Mechanical Treatment of Helium-Hydride . 121 comes Li+ as R 0, and in this limiting case the wave function is again approximated by with =43/16.Hence c/R should converge to (c/R) =27/32as R , whileit shouldtends to (c/R)=43/32 as R 0.The corresponding values in ease of H2 molecule are 0.500 and 0.844 respectively. Now, at the equilibrium nuclear distance R=1.40, James and Coolidge obtained their best results using c, ( in their notation) 0.75 or 0.875,which gives c/Rat the equilibrium nuclear distance (c/R) =0.536or 0.625. As- summing that (c/R)0-(c/R)e:(c/R)e-(c/R) is nearly the same in these two molecules, we obtain (c/R) =0.895 or 1.025 for HeH+. Hence, if we use the equilibrium dis- tance R of HeH+ given by Coulson and Duncanson R=1.446, the appropriate value of c is found to be in the neighbourhood of 1.294~1.482. This result shows us that we may take c=1.375 as the value of c at the equilibrium distance, in agreement with the estimation described in the preceding paragraph. § 4. The calculation of matrix elements of H and unity between functions of type (1) is carried out in a similar manner as in the case of H2(4). These matrix elements are linear combinations of intergrals of the type Numerical values of these integrals have been obtained from: " Tables of Integrals useful for the Calculations of Molecular Energies" by Kotani, Amemiva and Simose(5). In evaluating X's for which p=-1 -1 or +1, Neumann expansion of 1/r12are used, and tiny are reduced to whose values for 2c=2.75 are given in the above-mentioned Table. Num rieal value of matrix elements are given in Table I. Functions we have taken are as follows: (4) James, Coolidge, loc. cit. (5) Kotani-Amemiya-Simose: Proc. Phys. Math. Soc. Japan 20 (1938), extra no. 122 Soroku TOH [Vol. 22 Table I. Numerical Values of Matrix-Elements 1940] Quantum-Mechanical Treatment of Helium-Hydride. 123 Firstly we take 10 functions 1, 2, ...., 10, omitting the relatively un- important 11 and 12 and solve 10-dimensional secular equations for R=1.30, 1.40 and 1.50. Since the energy of HeH+ at R= is the energy of the He atom in the normal state, whose, experimental value is 5.807 Ry, it is necessary to subtract 5.807 Ry from all energy values in order to obtain the dissociation energy ED (including zero point energy). The result is shown in Table II and in Fig. 2. Table II. Constructing parabola passing through these 3 points, the equili- 124 Soroku TOH. [vol. 22 briun nuclear separation is found to be nearly R=1 .346. Then we solved once more the 10-dimensional secular equation for this value of R, and found ED=0.0887 Ry or 1.20 e.V ., which lies very nearly (0.0002 Ry lower) on the above-mentioned parabola . For this value of R we solved also a 12-dimensional secular equation, taking all 12 functions given in (3). Addition of 2 functions 11 and 12increases the dissociation energy by 0.08 e.V.. This value is also given in Table II and is shown by in Fig . 2. The numerical values of the coefficients A1 of the " best" wave function =A1 1+ A2 22+ is .... given +A21 1 in Table III. From the parabola mentioned above we obtain immediately the vibration frequency: The zero point energy is then 1700 cm-1 or 0.23 e.V .. This value must be subtracted from the values of ED given in Table II, if ED is measured from the wroth vibra- tional state. § 5. In order to know the rapidity of convergence and to examine the relative importance of various terms , we solved also for R=1.346 several secular equations, which are obtained by taking fewer func- tion, as shown in Table TV. Judging from There energy values , it seems that we are fairly near to the convergence limit . The true dissociation energy ED would be 1940] Quantum-Mechanical Treatment of Helium-Hydride . 125 probably 1.4 e.V. or even somewhat lower than that. It is to be remarked that the function 10=e is very effective in lowering the Table IV. energy. One of the principal causes why Coulson-Duncanson have ob- tained so small a value as 0.32 e.V. as ED is that our 10 is missing in their wave function. Moreover, the inspection of the coefficients of the "best" wave -function given in Table III shows that functions, which involve powers of 1, 2 are in general important, while e- enters into T with a very small coefficient. The negative sign of A2 and A10 means that the charge cloud is concentrated more in the neigh- bourhood of He nucleus, as of course should be*. The " best " value of Tr, which we have calculated, that is, 1.28 e.V. is the highest one hitherto obtained. The fundamental vibration frequency =3400 cm-1 is higher than that obtained by Beach and is very close to Coulson-Duncanson's value 3380 cm-1. Our equilibrium internuclear distance 1.346 is considerably smaller than the values calculated by these authors. This may partly he explained if we notice that the atomic orbital method gives larger equilibrium internuclear distance than the variation in the general method. In fact, we have found the lowest energy at R=1.37 with function (2). Finally, we solved also 4- and 5-dimensional secular equation with functions 1, 2, 3, 4, 5 and 6 for R=1.50. The result is shown in Table V. The first two values 5.8136 Ry and 5.8146 Ry agree with the corresponding values obtained by Coulson and Dnncanson. The fact * This corresponds to the fact that in (2) the sign of k is negative.
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