3080
J. Phys. Chem. 1986, 90, 3080-3088
Addltivtty of Vlbratlonal Zero-Point Energy Takao Oi,+ Anthony Popowicz, and Takanobu Ishida* Department of Chemistry, State University of New York, Stony Brook, New York 1 1 794 (Received: October 14, 1985; In Final Form: February 10, 1986)
The vibrational zero-point energy is expressible as a sum of contributions from valence coordinates and small correction terms due to intercoordinate interactions. Theoretical foundation for the additivity has been established on the basis of the orthogonal expansion for the zero-pint energy. The principle has been demonstrated by detailed analyses of various terms of the expansion for n-alkanes and n-alkyl chlorides. A semiempirical procedure for evaluating the additivity parameters has been devised and demonstrated for n-alkanes, C,-C,,, and n-alkyl chlorides, C1-C6. The parameters thus obtained yield a precision on the order of 0.01% for the zero-point energies of these molecules.
Introduction In the statistical mechanical formulations for the thermodynamic energy functions, including those for the transition-state theory, the terms of the zero-point energy (ZPE) are the most significant ones at ordinary temperatures. At the near- and subambient temperatures the ZPE term alone is often a good approximation for some thermodynamic functions. However, the vibrational ZPE, defined as a product of ( h / 2 ) and the frequency sum, or the sum of the zeroth-order frequencies (wi) over all internal degrees of freedom, is not known for the large majority of polyatomic molecules: Only for the simplest molecules are data on the anharmonicity constants (xu, xijk,...) available, and even the fundamental frequencies (vi) are not completely known for most of the molecules. In the absence of such information, the common practice has been to use Cvi in the place of Cui in these calculations. To the first order in the anharmonicity correction the error that one introduces in this procedure for the frequency sum is (3/4)Zx,,, which is on the order of 1% of Cui for water and smaller for non-hydrogenous molecules. The terms of xij ( i # j ) in the expression for the vibrational energy term cancel out between Cui and Cvi. Since the error due to Cxiiis negative for all stable molecules, one expects some cancellation between the error terms when a difference in ZPEs is taken. It is on this background that attempts for empirical ZPE additivity rules have been made in the past on the basis of observed fundamentals. The concept of additivity for the vibrational zero-point energy is an old one. In 1937 Pitzer assigned an empirical value for the vibrational frequency of each type of bond stretching and bending motion as the basis for computation of thermodynamic functions of gaseous hydrocarbon^.'-^ In 1948 Cottrel14 pointed out a constancy of the incremental increases in the ZPE with successive additions of methylene groups to the n-paraffin chain. The constant value of 17.7 kcal/mol CHI was later used by Pitzer and CatalanoS in their empirical linear expression for the ZPEs of paraffins, which gave precisions ranging between 0.1% and 1% for n I5 in CnH2n+2. Between 1952 and 1965 Bernstein published several papersb" on multivariable linear least-squares formulas for the ZPEs of halomethanes, haloethylenes, and various isotopic homologous series of methane, ethylene, and benzene. He took into account not only contributions of interactions between bonded atoms but also those between the atoms which are two bonds and three bonds away. His methods generally yielded precisions between 0.1% and a few percent. Fujimoto and Shingul* empirically fitted a three-parameter linear expression to experimental data on the ZPEs of normal and branched paraffins and a five-parameter expression to those of olefins. The parameters for the paraffins are contributions of each C-H bond, each C-C bond, and an end-of-chain correction. Their 'Present address: Department of Chemistry, Sophia University, Tokyo
102, Japan.
0022-3654/86/2090-3080$01.50/0
formula generally led to precisions similar to that obtained by Bernstein. The precisions quoted for these attempts are relative to ZPE values obtained from experimentally observed fundamentals and therefore lack internal and relative consistency. These empirically derived additivity rules for the ZPE can be given a theoretical foundation in view of the recently developed orthogonal a p p r o ~ i m a t i o n ' ~for ' ~the ZPE. In this paper we will show how these rules with various levels of sophistication could be derived as special cases of the more general rule. A new empirical method will then be derived from the general rule. The derivations and expressions presented in this paper are strictly applicable only for the systems of harmonic oscillators. In the applications of these additivity methods, however, only the fundamental frequencies are of practical value. Thus, practical F matrices have been fitted to observed fundamentals, and vibrational eigenvalues, normal frequencies, and the ZPEs have been obtained from such F matrices, as if they were for the systems of harmonic oscillators. The ZPEs thus calculated contain intrinsic errors, but their magnitudes are unknown. We believe that the methods presented here are useful in demonstrating the principle and that the tables of empirical additivity parameters derived here are useful since it is the best practical alternative in the absence of the anharmonicity data. Theory The vibrational ZPE of a system of harmonic oscillators can be expressed as a truncated power seriesI3 in the vibrational eigenvalues Xi on the basis of Lanczo's 7-method:l6 ZPE
h J
I
-CXi'/2 2 i
(1)
(1) Pitzer, K. S. J. Chem. Phys. 1937, 5, 473. (2) Pitzer, K. S. J . Chem. Phys. 1940, 8, 711. (3) Pitzer, K. S. Chem. Reu. 1940, 27, 39. (4) Cottrell, T. L. J . Chem. SOC.1948, 1448. (5) Pitzer, K. S.; Catalano, E. J. Am. Chem. SOC.1956, 78, 4844. (6) Bernstein, H. J. J . Chem. Phys. 1952, 20, 263, 1328. (7) Bernstein, H. J.; Pullin, A. D. E. J . Chem. Phys. 1953, 22, 2188. (8) Evans, J. C.; Bernstein, H. J. Can. J . Chem. 1955, 33, 1171. (9) Bernstein, H. J. Can. J . Chem. 1956, 34, 617. (10) Bernstein, H. J. J . Chem. Phys. 1956, 24, 91 1. (1 1) Bernstein, H. J. J. Phys. Chem. 1965, 69, 1550. (12) Fujimoto, T.; Shingu, H. J. Chem. SOC.Jpn. 1962, 83, 19. (13) Oi, T.; Ishida, T. J. Phys. Chem. 1983, 87, 1067. (14) Oi, T.; Ishida, T. J . Phys. Chem. 1984, 88, 1507. (15) Oi, T.; Ishida, T. J. Phys. Chem. 1984, 88, 2057. (16) Lanczos, C. Applied Analysis; Prentice Hall: Englewood, NJ, 1956.
0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 14, 1986 3081
Additivity of Vibrational Zero-Point Energy The summations in eq 1 and 2 are taken over the internal degrees of freedom,f, and over the order of the expansion, N , respectively. T r ( H J ) is the trace of the J t h power matrix of H FG, with an understanding that T r (P) = n, and F and G are the vibrational force constant matrix and the effective inverse mass matrix, respectively. The coefficients bJ are simple functions of two approximation parameters X, and t,I3and have a dimension of X1/z/XJ; Xo is a reference eigenvalue chosen for the purpose of eigenvalue normalization, and [ is the range of the normalized eigenvalues Xi/housed in the expansion. The optimum values of the approximation parameters were obtained’) for the N = 2 expansion by minimizing the sum of the squares of the approximation errors weighted by functions of the form, w(X)aXk. Following the notation used by Wolfsbergl’ and Bigeleisen,’* and Go = (gi;), where let us define diagonal matrices, Po = = fiJaOand g: = giJaij,in which 6, is the Kronecker delta, and f i j and gij are the (id) elements of the F and G matrices, respectively. Then, the ZPE of the imaginary molecular system of completely uncoupled oscillators, (ZPE)o, is given by
fiy
hf
(ZPE)o = -Ccf;igii)l/z 2 i
(3)
(4) where 6 = FOGo. If we choose to use the same approximation parameters for both ZPE and (ZPE)o, we have, for N = 2 ZPE =
hc Tzui i f
h f - [ ~ c f ; ~ i g i i )+ ’ / bl * ATr (H) 2 i
=
Cij1 = 2&Jj
Ci?( 1) is the second-order correction attributable to interaction between ith and j t h coordinates due to nonvanishingf;i Cij2(1
s
(5)
where Cpi is the sum of internal frequencies in wavenumbers, which we will call the frequency sum, FS, and ATr (H) = T r (H) - T r (Ho)
(11)
C i t ( 2 )is the second-order correction attributable to interaction between ith and j t h coordinates due to nonvanishing gij and f i j Ci?(2) = 4giJjkihi + gjhj)
(12)
Ci;(3) is the second-order correction attributable to interaction elements gij and
fili
Ci?(3) = 2gij2fij2
(13)
C J ( 4 ) is the second-order correction attributable to interaction between ith and j t h coordinates due to nonvanishing gij Ci?(4) = 2gij%&j
(14)
CO$(1) is the second-order correction attributable to simultaneous interactions of ith coordinate with j t h and kth coordinates due to nonvanishing gjk, J j , and f i k cijk2(1) =
4gilgjddk
(15)
c0k2(2)is the second-order correction attributable to simultaneous interactions of ith coordinate with j t h and kth coordinates due to nonvanishing fik, gij, and gik (16)
c0k2(3) is the second-order correction attributable to simultaneous interactions of ith coordinate with j t h and kth coordinates due to nonvanishing gij, gikr f j , and f i k
Cijk’(3) = 4gijgidLk
(17)
c i j k f ( 1) is the second-order correction attributable to nonvanishing kinetic interactions, gij and gkl, due to nonvanishing fi, and f j k
Cijkt(1) =
4gijgkh&k
(18)
Cvu2(2)is the second-order correction attributable to nonvanishing kinetic interactions, gij and gkl, due to nonvanishing f i k and fj,
and ATr (H2) = T r (H2) - T r (Hoz) = CCCnfklgikgjl
cijk?(2) = 4gijgklf&I (7)
i J k l
For eq 7 the quadrupole sum is taken over all possible combinations of the indices i, j , k, and 1 allowing duplications, with the exception of the cases i = j = k = 1. The term of Cicf;igii)1/2 is the zeroth order term, to which the ith coordinate contributes cf;igii)l/2.The terms of ATr (H) and ATr (Hz) are the first- and second-order correction terms, respectively. Various terms of ATr (Hz)can be classified into several groups on the basis of how many and which of the four indices are identical. Thus, eq 5 is rewritten as follows. ZPE
kiigjfi?
cijk2(2) = 4gijgid&k
+ b2 ATr (Hz)]
(10)
4
h
,[zC; i
+ blCCCijl + bzCC C,‘(m) + i
