Note on the Calculation of Zero-Point Energies and Isotopic Zero-Point Energy Differences by a Taylor's Series Expansion* By J. BIGELEISEN, R. E. WESTON, JR., and M. WOLFSBERG Chemistry Department, Brookhaven National Laboratory, Upton, L. I., New York (Z. Naturforschg. 18 a, 210—215 [1963] ; eingegangen am 26. November 1962) KLAUS CLUSIUS zum 60. Geburtstag gewidmet The series expansion for the zero-point energy of a system of harmonic oscillators derived by BIGELEISEN and GOLDSTEIN has been examined in detail. The convergence properties have been tested as a function of the reduced frequency, and the error produced in truncating the series has been found. Calculations of zero-point energy sums by solving the secular equations for isotopically sub- stituted series of hydrogen, methane, ethylene, ethane, and benzene molecules have been made and the results are discussed in terms of the first- and second-order isotope sum rules. In the preceding paper 1, it has been shown that valent to a knowledge of the zero-point energy £0 . the reduced harmonic zero-point energy of an n- The practical usefulness of (1) arises from the fact atomic molecule 3n—6 n 2n 3n—6 that 2Ä xk (or 2 co* ) can be obtained from the £q=I 2]eOft/a>0 secular determinant in terms of force constants and atomic masses by means of the sum rules 2 without (where co^. refers to the eigenfrequencies of the explicit calculation of the individual frequencies. molecule and co0 is a characteristic frequency to be chosen) can be expressed in terms of a power series I. Individual Frequencies 3n—6 3 re-6 2 xk 2£n = 2 ' 2 Figures 1 and 2 show the convergence of the series 3n-G oo (1) for an individual frequency (i.e. 3n —6 = 1) (1) as a function of x. The convergence criterion sets + 2 2 22/>-l (p_i) 1 pi k p=i limits on x of 0 x 2. We see from the figures where xk = (coÄ/co0)2. This series is absolutely con- that the rapidity and the type of the convergence vergent for \xk — 1 j 1, k — 1,..., 3/1 — 6. The do show a dependence on i, It is to be observed, purpose of this paper is to examine the convergence however, that if we restrict our attention to the properties of this series for various values of x, to range 0.25 ^ x ^ 2, the maximum error made at determine the errors produced by truncating the p = 2 in the summation is of the order of 10%. We series at various terms, and to show the behavior of should consider this in the light of the fact that we equation (1) when some representative molecules want to use the series (1) to calculate the zero-point are considered. Furthermore, the effect of changes energy of a polyatomic molecule (i. e. several fre- in certain vibrational force constants on the terms quencies) with a single value of co0 . (It is obvious, of equation (1) will be discussed. Particular interest of course, that the sum rules can be used in conjunc- is attached to the calculation of zero-point energy tion with series (1) only if a single characteristic differences between isotopically substituted mole- frequency, to0 , is employed 1.) Thus for a molecular cules. In the present context, series (1) has been species having frequencies which differ by as much evaluated by means of calculated values of xki/2 as a factor of 2.83 ( i. e. 1/2/1/0.25), it is possible (i. e. frequencies). From the practical point of view to pick a value of co0 such that the zero-point energy this would obviously be completely useless because is given by the first three terms of series (1) to a knowledge of xk112 (k = 1,..., 3 n — 6) is equi- better than 10%. For organic molecules containing * Research carried out under the auspices of the U.S. Atomic 2 J. BIGELEISEN, "Proceedings of the International Symposium Energy Commission. on Isotope Separation", edited by J. KISTEMAKER, J. BIGEL- 1 J. BIGELEISEN and P. GOLDSTEIN, Z. Naturforschg. 18 a, 205 EISEN and A. O. C. NIER, North-Holland Publ. Co., Amster- [1963]. dam 1958, pp. 122-125. Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschung This work has been digitalized and published in 2013 by Verlag Zeitschrift in Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung der für Naturforschung in cooperation with the Max Planck Society for the Wissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht: Advancement of Science under a Creative Commons Attribution Creative Commons Namensnennung 4.0 Lizenz. 4.0 International License. x Fig. 1. Fig. 2. Figs. 1 and 2. Ratio of calculated frequency to actual frequency as a function of x (x=co2/a>02) at various terms in series expansion (equation 1). carbon-hydrogen bonds, the maximum frequencies + 800 are about 3000 cm"1. Then with a)0 = 3000/V2 , we are assured of the above 10% convergence if there are no frequencies in the molecule less than 1000 + 700 cm-1. The 10% figure actually applies only to the smaller frequencies in the above range and there + 600 may be some cancellation of errors. Hence, we would expect to be able to pick a value of o)0 such 7 +500 5 that the series (1) converges to the zero-point o energy to better than 10% for the first three terms j +400 < over a much larger range of frequencies than z> t- 1000 cm-1 to 3000 cm-1. The contribution of ex- o tremely low frequencies may be estimated quite in- < +300 3 correctly by the series (1) but such frequencies tend o to contribute proportionally rather little to the total o< +200 zero-point energy. + 100 These considerations are demonstrated in Fig. 3, in which the differences between actual frequencies and those calculated from equation (1) are plotted. 0 With a choice of co0 = 2000 cm-1 (corresponding approximately to the value needed in a molecule -100 with a C —H bond), frequencies in the range 1000 to 2800 cm-1 are in error by no more than 100 Fig. 3. Error in calculated frequency with o>0 = 2000 cm cm-1 when the first three terms of the series are various terms in the series. used. The errors become larger for frequencies tions with frequencies below 1000 cm-1 are usually below 1000 cm"1. skeletal vibrations which are relatively insensitive Fig. 4 shows the error which results when equa- to isotopic substitution of a hydrogen atom. tion (1) is used to approximate the change in fre- It should be pointed out that in the comparison quency produced by isotopic substitution. The or- of frequencies of different isotopic species, a some- dinate is the error in the difference co1 — io2, where what more stringent requirement is placed on the we have used the ratio coj/cog = 21/2, i. e. the maxi- choice of co0 . In order to take advantage of the con- mum isotope effect to be expected in a molecule venient form of the isotopic sum rules 1, one must where a hydrogen atom has been replaced by deu- use the same w0 for the different isotopic mole- terium. The maximum error (for p ^ 2) is found cules. The use of the minimum value allowed by when co2«i400 cm-1. Fortunately, molecular vibra- the high frequencies of the light isotopic species will lead to smaller values of a)/co0 in the heavy isotopic species, and thus to less rapid convergence of the series. The behavior of series (1) for an actual molecule is shown in Table 1, which gives results obtained for the H2, HD, and D2 molecules. This system is expected to show the largest deviation from the linear zero-point energy rule (law of the mean), i. e. the largest difference between co(H2) - co(HD) and CO(HD) — CO(D2). It is seen that convergence is ex- tremely rapid for the three values of co0 which were used. II. Polyatomic Molecules Tables 2 through 6 contain the zero-point energies and zero-point energy differences for several iso- topically substituted series of larger molecules. These values were obtained from exact solutions of the secular determinants for the vibrational frequencies. In addition, some comparisons are made with zero- point energies calculated with the series expansion (1). p Methane. — The zero-point energies of CH4, Fig. 4. Error in frequency difference, co1 — a)2, with (o1fco2 -1 = ]/2 and co0 = 2000 cm at various terms in the series. CH3D, CH2D2, CHDG, and CD4 were calculated P = 1 2 3 4 5 6 Exact 640.54 581.75 592.13 589.79 590.35 590.22 590.28 Wo = OJH2 — ®HD COHD — COD2 640.54 690.05 698.08 699.71 700.06 700.17 700.19 3788.19b 0 2 COHD — (TT>H2 + COD2) 108.30 105.96 109.92 109.70 109.95 109.90 COH2 — COHD 645.40 580.36 592.58 589.72 590.47 590.26 590.28 CÜ0 = 645.40 691.12 698.38 699.77 700.11 700.19 700.19 3760.00 c WHD — COD2 2 COHD — (C«H2 + C«D2) 0 110.77 105.81 110.06 109.64 109.93 109.90 co0 = I 2 OJHD — 0 106.00 106.00 109.66 109.66 109.90 109.90 3815.61d (®H2 + G>D2) a b c 1 d Based on calculations by P.GOLDSTEIN. a>0= (a>H2 + WD2 + 2 o>hd)/4.