J. Phys. Chem. 1963, 87, 1067-1073
1067
Correlatlon of Zero-Point Energy with Molecular Structure and Molecular Forces. 1. Development of the Approximation' Takao 01 and Takanobu Ishida' Lbpattment of Chemistry, State Universw of New Y&, I n Final Form: November 4, 1982)
Stony Brook, New York 11794 (Received:August 23, 1982;
An approximation formula for the zero-point energy (ZPE) has been developed on the basis of Lanci" r method, in which the ZPE has been expressed in terms of the traces of positive integral powers of the FG matrix. It
requires two approximationparameters, Le., a normalization reference point in a domain of vibrational eigenvalues and a range for the purpose of expansion. These parameters have been determined for two special cases as well aa for general situation at various values of a weighting function parameter. The approximation method has been tested on water, carbon dioxide, formaldehyde, and methane. The relative errors are 3% or less for the molecules examined, and the best choice of the parameters moderately depends on the frequency distribution.
Introduction Any mathematical function which is continuous and bound in a given domain of an independent variable is expressible as a linear combination of an orthonormal set of functions defined in the same domain. Thus, the reduced partition function ratio for a pair of isotopic mol~ been approximated by means of ecules, In ( s / s ? ~ ,has various Jacobi polynomial^^*^ with the aid of Lanczo's 7 method5 which allows an extension of the domain to suit one's need. Since the resulting formula can be rewritten into the form of a linear combination of the traces of increasing powers of the FG matrix (or those of the GF matrix), the method and the ensuing improvement6in the approximation have provided explicit relationships between In (s/s?f and the elements of the F and G matrices, thus affording a means of elucidating the correlation between the isotope effect, molecular forces, and molecular structures.G13 The MINIMAX method developed by Jancsb, NBmeth, and Gellai14J5also led to a similar expression of In (s/s ?f in terms of the traces and, more recently, Lee and BigeleisenI6obtained an improved set of expansion coefficients in their WINIMAX approximation. Concurrently to the studies of these methods, Wolfsberg"J8 developed an
ingenious second-order perturbation method for expressing
In (s/s?fin terms of the elements of the F and G matrices. Both methods have been instrumental in circumventing the black box, i.e., the computerized solution of vibrational secular problems, and providing a direct link between molecular structure and molecular forces and the thermodynamics of isotope chemistry. In this paper we report on a development of a new explicit relation between molecular structure and molecular forces and the zero-point energy (ZPE) and its numerical tests on the W E of water, carbon dioxide, formaldehyde, and methane. The only earlier attempt of the same goal has been, to the best of our knowledge, the one by Bigeleisen and Gold~tein'~ and the perturbation calculation by Wolfsberg.20 The former authors expanded the molecular sum of frequencies, wi, in a Taylor series of even powers of ai's around an arbitrarily chosen point wo along the frequency spectrum:
where Xi = 4?r2vt = 47r2c2wt, Xo = 4?r2c2wO2,and (-1)p+'(2p - 2)! cp
(1) Research supported by the Office of Basic Energy Sciences, U S .
Department of Energy, Contract No. DE-AC02-80ER10612. (2) Bigeleisen, J.; Mayer, M. G. J. Chem. Phys. 1947, 15, 261. (3) Bigeleisen, J.; Ishida, T. J . Chem. Phy8. 1968, 48,1311. (4) Ishida, T.; Spindel, W.; Bigeleisen, J. Adu. Chem. Ser. 1969, No. 89, 192. (5) Lanczos, C. 'Applied Analysis";Prentice Hall: Englewood Cliffs, NJ. 1956. (6) Bigeleisen, J.; Lee, M. W.; Ishida, T.; Bigeleisen, P. E. J. Chem. Phys. 1978,68, 3501. (7) Bieeleisen. J.: hhida. T.: hindel. W. R o c . Natl. Acad. Sci. U.S.A. 1970; 67,-113. (8) Bigeleisen, J.; hhida, T.;Spindel,W. J. Chem. Phys. 1971,55,5021. (9) Bigeleisen, J.; Ishida, T. J. Am. Chem. SOC. 1973, 95, 6155. (10) Bigeleisen, J.; Ishida, T. J . Chem. Phys. 1975, 62, 80. (11) Bigeleisen, J.; Hom, R. C.; Ishida, T. J. Chem. Phys. 1976, 64, 3303. (12) Ishida, T.; Bigeleisen, J. J. Chem. Phys. 1976, 64, 4775. (13) Bigeleisen, J.; Ishida, T.; Lee, M. W. J. Chem. Phys. 1981, 74, 1799. (14) Jan&, G.; NBmeth, G. Gellai, B. Chem. Phys. Lett. 1970, 7,314. (15) N6meth, G.; Gellai, B.; Jancab, G. J. Chem. Phys. 1971,54,1701. (16) Lee, M. W.; Bigeleisen, J. J. Chem. Phys. 1978, 68, 3496. (17) Singh, G.; Wolfsberg, M. J. Chem. Phys. 1975,62, 4165. (18) Harvie, C. E.; Bopp, P.; Wolfsbrg, M. J. Chem. Phys. 1980, 72, 6349. .
I
I
I
-
I
0022-3654/83/2087-1067$01 SO10
=
22P'(p
- I)!
Although the series absolutely converges when Xi f A,, 5 2 for all i, the convergence is rather slow, and there is no set guide for finding the best value of A,,. The method we will present here is based on the formula of Lanczos? obtained by an expansion of function x1/2 in terms of orthogonal polynomials: M
=
c amxm
ma0
0 Ix I1
(3)
where M is the order of expansion, and (4)
in which CE is the coefficient of the mth order term in an orthogonal polynomial of order M defined in 0 I x I1. (19) Bigeleisen, J.; Goldstein, P. Z . Naturforsch. A 1963, 18, 205. (20) Wolfsberg, M. 2.Naturforsch. A 1963, 18, 216.
0 1983 American Chemical Society
1068
Oi and Ishida
The Journal of Physical Chemistry, Vol. 87, No. 6, 1983
Mathematical Procedures and Results Let the normal frequency spectrum of an N-atomic molecule be represented by the following ascending sequence of vibrational eigenvalues: X2,
..*, Xi,
*e*,
An (= Amax)
**.,
xn
(= XmaJ
(6)
where xi
Xmax
= AJXO
= Xn =
Amax/&
(10) now becomes5
= -1
When M = 2, eq 10 becomes h" 2i
ZPE = -C[bo(Xo,S) + bi(Ao,t)hi+ where bo = aoXo112[2/D bl = alXo-1/2[/D
(8)
Then, the ZPE can be approximated by
CY,
TI*(X)
(7)
At this point, A, is completely arbitrary so that some of the xi's may be greater than unity. In order to take advantage of eq 3, we define the range of expansion, [, such that (9) Xmax I 5
where
+ 2x T~*(x = )1 - 8~ + 8 x 2
(5)
in which degenerate eigenvalues have been counted separately, so that n = 3N - 6 (or 3N - 5). Corresponding to the sequence 5, we may think of a normalized sequence, that is, normalized to a certain value Xo, which would be determined later: ~ 1 ~, 2 * *,* , xi,
although any other orthogonal set of polynomials defined in 0 5 x I1would have been acceDtable. The first three Tn*(x) are as follows: To*(x) = 1
b2 = a2Xo-31z/D
in which
ao[2+ al[ + a2 a. = -1 al = -8 a2 = 8/3 D
(21)
In passing we note that D = 0 at [ = 0.320494. As it happened, however, all optimum values of [ we obtained were greater than this value, making D negative under all situations. The best values of & and [ would be those that minimize the error
S
Cw(Xi)A? i
(22)
where w(Xi) is a weighting function, and Ai
n
CX$ = Tr (H") i
(12)
where H FG, and Tr (A) is the trace of a square matrix A, the first two of which being (13) Tr (H2) = CCCCfikfjgkjgli i l k 1
(23)
Obviously, such optimum values of & and [ depend on the molecular spectrum and, thus, on the weighting function. In the absence of an a priori knowledge of such frequency distribution, we used a functional form
Further, we have
= Tr [(FG)"]
bo(Ao,[)+ bl(Xo,[)Xi + bZ(&,t)X? - Xi1/'
(14)
Thus,eq 10 represents an explicit relationship between the ZPE and the molecular structural parameters and the internal vibrational forces. However, the two parameters, & and [, that we have introduced remain to be determined. Based on our past e~perience~J*'~ in which the studies of correlation between the isotope effect and the elements of F and G matrices had been carried out, we decided at the very beginning that we would not attempt to make correlation studies involving H3 and the higher terms. That is, we limited ourselves to the A4 = 2 approximations only, because the individual terms included in the higher-order traces are too numerous and too hard to keep track of to make the resulting interpretation physically meaningful. In the best approximation6for In (s/s?f based on the method of the moments of vibrational eigenvalues, omission of H3and higher terms yields errors which are within 1% of the exact values for deuterium substitution and 2.6% for isotopic substitution of other elements. In addition, the presentation that follows is limited to the shifted Chebyshev polynomials of the first kind, Tn*(x),
w(Xi) =
AX!
(24)
in which A is a normalization constant, and k a variable parameter. For various preset values of k , the best values of ([,Ao) were determined by minimizing S by simultaneously solving as/aXo = o (254
asla[
=
o
(25b)
For the sake of simplification of the mathematics, we treated S defined by eq 22 as a continuous function in the domain, 0 I X I Amax. Consequently, the normalization constant of eq 24 became A = - k- -+ l k+ 1 Xmax
and S of eq 22 became
Besides a general treatment, two special cases were also studied. In the first special case, we set Xo =, ,A i.e., all eigenvalues were normalized to the largest one. Then, xi I1for all i, and the requirement for the range of expansion is f 1 1. In the second special case, we set f = xm, = Am,/&, so that the requirement on the value of x for the validity of the Lanczos formula is automatically satisfied fo all eigenvalues. This [ represents the smallest
Correlation of ZPE with Molecular Forces
The Journal of Physical Chemistry, Vol. 87, No. 6, 1983
allowable value of 5 for the given A@ Case A. A. = =.A, For the special case in which A. is set equal to , , ,A the only parametric variable, 5, was determined by solving eq 25b which led to g(5) = 0
TABLE I: Best Values of as a Function of k for Case A, h, = ,,,A (Three Roots of Eq 28)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 10.0 15.0 20.0
(28)
where g(x) =
2a22 - -@a$ 5+k
+ a l ) - -42a1a2 (3ag2 +k
+ alx - a2) +
+
2(aI2 2aoa2) ( a g 2- a2)x + 3+k 2a1 (aox2- a2)(a,g2+ alx + a2) 72 + k 2aoa1 -(aG2 - alx - 3a2)x22+k 2a0 (alx 2a2)(a,p2+ alx + a2)x + Y2 + k 2a02 + 2a2)x3 (29) -(alx l + k
+
A rearrangement of the terms of g ( x ) and application of Descartes’ rule of signs show that the equation possesses three or one positive real roots. In fact, three positive roots were found, for every value of k examined, by means of the algorithm of Newton’s successive linear approximation. The limiting values of three roots were 5 N 0.85, 1.83, and 2.15 a t k = 0, and 5 N 1.17, 2.43, and 6.83 at k = 03 (see footnote a of Table I). A short list of these roots is given in Table I. Precision of approximations for the ZPE obtainable with these values of 5 and Xo =, , A will be discussed later. Case B. 5 = &/A,,. For the special case of 5 = &I&, the only variable, E, was determined by solving aslag = 0, which led to GI([) G2(O = 0
(30)
where G l ( x ) = a G 2 - alx - 3a2 G ~ ( xE) aZ2 5+k
(-
+-42ala2 +k+
a12+ 2aoa2 3+k
(31)
-)..
2aoal a +2+k +l + k
1069
ma
0.849 1 2 0.876 48 0.898 8 3 0.918 0 3 0.934 77 0.949 49 0.962 53 0.974 1 5 0.984 57 0.993 96 1.002 45 1.057 29 1.085 31 1.102 31 1.171 57
1.834 37 1.665 32 1.706 55 1.759 90 1.810 41 1.855 61 1.895 49 1.930 65 1.961 72 1.989 31 2.013 94 2.164 22 2.235 18 2.276 35 2.430 50
2.147 21 3.179 50 3.752 41 4.160 45 4.470 11 4.714 30 4.912 21 5.076 0 3 5.213 94 5.331 67 5.433 38 5.997 85 6.237 23 6.369 54 6.828 43
The value of k higher than l o 6 , beyond which the roots are within of their asymptotic limits. a
TABLE 11: Best Values of 6 as a Function of ka for Case B, E = h m a x / h ,
k 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0
EW)
E(b1) 0.867 17 0.897 9 3 0.926 22 0.95022 0.970 20 0.98672 1.000 30 1.01140 1.020 39 1.027 6 3 1.033 38 1.04140 1.04602 1.048 36 1.04921
1.746 08
E(b3) 19.836 36
1.566 35
20.751 94
1.463 61
21.227 86
1.404 27
21.472 68
1.368 67
21.604 78
1.347 1 5
21.678 1 2
a The three roots of G , ( x ) = 0 ( c f . eq 32) yield identical ZPE approximations. Values of g(b2) and E(b3) are shown at several k values to facilitate comparison. T w o
roots of G , ( x )= 0 ( c f . eq 31), t(b4) = 4 - 2(2)’” and ~ ( b 5=) 4 t 2(2)’/, for all k , do lead t o different approximations.
the approximation, eq 16, appears as a single factor, 53/2/D, which ratio also satisfies G2(5)= 0. Numerical results of the approximations obtained for this special case will be discussed later. Case C. General. In the general case eq 25a becomes (33)
(32)
The roots of Gl(5) = 0 are 4 f 2(2)’12. In agreement with Descartes’ rule of signs three positive roots of G2(5)= 0 were found for every k value we studied. They range from 5 N 0.9, 1.7, and 19.8 at k = 0 to 5 N 1.0, 1.5, and 21.5 at k = a. Representative values of the roots have been tabulated in Table 11. It is generally expected that, for a given set of discrete frequencies, the two roots of Gl(x) = 0 will yield two approximations which are independent of the k value, inasmuch as the function G l ( x ) and the parameters involved in the ZPE approximation, eq 16, do not contain k. In addition, it is seen that the three roots of G2(0= 0 will lead to ZPE approximations which are identical with each other, because the forms of bo, bl, and b2 in this special case are such that the 5 dependence of
a2)y5I2-
(U12
+ 2aoa2)x2y2 I 3+k
72
a0 a , ) ~ y-~/ ~ ( a G 2 + alx 72 + k
a1 ( a G 2 + alx +
U02 + a2)x2y1/2+ -x4 l + k
+ (35)
Similarly, eq 25b becomes G(E,tl) = 0
where
(36)
1070
The Journal of Physical Chemistty, Vol. 87,
No. 6, 1983
G(x,y) 2a22 2W2 -(2aox + al)y4- -(3a0x2+ alx - a2)y3+ 5+k 4+k
Oi and Ishida
TABLE 111: Best Values of 6 and q as a Function of 12 for Case C, General 0.172663 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 5.0 7.0 10.0 15.0 20.0
2 ( q 2 + 2a a O 2) (aox2- a,)xy2 + 3+k
cab
2a02 -(ulx l + k
+ 2a2)x3 (37)
Equations 33 and 36 were simultaneously solved by the two-dimensional Newtonian algorithm:
2.438 38 3.258 79 3.856 21 4.264 9 1 4.570 27 4.808 92 5.001 20 5.159 66 5.292 64 5.405 88 5.788 69 6.042 37 6.269 62 6.394 96 6.828 4 3
1.443 47 1.319 60 1.246 1 7 11.566 8 9 3.737 9 6 1.202 28 1.172 1 5 9.982 92 2.560 34 1.149 9 8 1.132 9 3 9.202 11 2.084 57 1.119 39 1.108 36 8.732 96 1.828 81 1.099 20 1.069 74 1.051 42 1.035 75 1.027 40 1.000 00
( t ( c l ) ,q(c1)) is the smaller root of eq 33 and 36. The second root is given for some values of k to facilitate various numerical comparisons. For each k value eq 40 and 41 are satisfied, and [[(cl), q ( c l ) ] and [t(c2), q ( c 2 ) ] yield an identical ZPE approximation. Also, note that A, = Amax/q. The value of k higher than lo6, beyond which the roots are within 10.' of their asymptotic limits. 10
5 r l
$
where ti and qi are, respectively, the ith approximations for and q in the iterative process, and F,, F,,, G,, and G , are (dF/dS), (dF/dq),(dG/dS),and (dG/dq),respectively. For physical reasons no attempts were made to find the roots other than the two presented here. A sample list of these roots is found in Table 111. We note that the smallest k value for which these roots exist is 0.172 663. At every value of It we studied, the ratio of S and q was found to be identical for the two positive roots, and they were found to yield an identical ZPE approximation. The coefficients of the ZPE approximation, bo, b,, and b2, depend on the factors, S2q-1/2/D,Sq1J2/D,and q3I2/D,respectively. It can be shown that a requirement that these factors be constants and independent of individual values of 5 and q explains both facts found numerically. Let
D = y1&-1/2 = y2Sq1/2 = y3q3/2
(40)
where yl, y2, and 73are the constants. These would depend on k . First, the equation is consistent with the fact that the ratio, S / q , is the same for all roots at a given k:
S/v
= 72/71 =
73/72
(41)
This also implies that 722 = 7 1 7 3 . Secondly, the ZPE approximation becomes independent of individual values of 5 and q, since the only t- and q-dependent factors of bo, bl, and b2 are now constants, yl, y2,and y3,respectively. Thirdly it can be shown that eq 40 simply represents a transformation of variables for eq 35 and 37 and does not impose any constraint on the system of equations and, consequently, that the general solution of eq 33 and 36 obtained without regards to eq 40 leads to the roots (5,~) which turn out to satisfy eq 40. For instance, if eq 40 is used, eq 35 can be transformed into an order-four polynomial of a ratio, say, (y1/y2)with coefficients which depend on ao,al, a2,k , and y2,i.e., FT(yl/72);y2].Descartes' rule predicts three or one positive roots for (yl/y2) in F' = 0. Similarly, eq 37 can be transformed into a quadratic equation of x (or 6, the only difference being that is the
Y
Y K 0
E
4
> '
Y
L
O
0.5 Y
Q
.-c>
0
Q
o?
0.1
0
1
2
3
5
4
6
7
8
9
IO
k Flgure 1. Relative errors in the ZPE approximations for the model system, (S/&)"*, as a function of k (cf. eq 42): a/ (j = 1, 2, 3) refer to [(a,) of case A. b, (j = 1, 4) refer to [(b)) of case B. [(b2) and E(b3) yield plots identical with that of [(bl). c, refer to C;(cl). f(c2) gives the same results as that of [(cl).
root value for x ) with coefficients which are polynomials of (yI/y2),i.e., Gqx;(yl/y2)]. An iterative process would solve the transformed equations for two unknowns, (y1/y2) and x : Due to eq 40 and its consequence, eq 41, a given set of (yl/ y2)and uniquely determine all other variables, i.e., yl, y2,y3,D, and 7. The fact that the two roots numerically found do satisfy eq 40 corresponds to the situation in the transformed set of equations that one of the possible roots, (rl/yz),of F' = 0 yields two real positive roots, hcuand 5(c2, in G' = 0 which happened to be the two roots of eq 33 and 36. In passing, we note that a requirement, y1 = y2 = y3, imposed on eq 40 does represent a constraint and leads to the special case B.
The Journal of Physical Chemistry, Vol. 87, No. 6, 1983
Correlation of ZPE with Molecular Forces
TABLE IV: Approximation Coefficients & j a and case A (A, = h,,,)
Pjb
case C (general)
case B ( t ; = h,,/h,)
j
bi
Pi
0 1 2
a ,E 2hmax1/zID a E hmax'l/z ID a ,Am& ' I 2 / D
a,t2/D alt;lD a,lD
1071
bj
PiC
bjc
Pj
a o ~ 3 ~ z h m a x 1 i z / DO , ~ ; ~ ~ ~ a/ o tD2 h m a x l / 2 9 -112 ID a I[ 312h Iiz/D alt;3Jz/D a E h max-1/2q ID U 1 5 3 / 2 h m , - 3 / 2 /D azE3/z/D azhm,- '/'q 312 / D
aotzq-liz/D alt; V " ~ / D a2q5/2/D
See eq 17-19 for the general definitions. To be used in eq 42 for the measure of the relative error. q = kmax/ho: and q t o be used for these coefficients are the "best values" obtained by solving eq 25a and/or eq 25b: See Tables 1-111, for example. TABLE VI: Approximations for the ZPE of Carbon Dioxide, 1zC160,[ C W = ~ 5018.7 cm-"]
TABLE V: Approximations for ZPE of Water, H160 [ x u i = 9425.5 cm-l] approximation for 2 w k
0.0
Aa Bb
0.173d
Cc
0.5
A
B C 1.0
A
B 2.0
C A
B C 5.0
A
B C a
cm-I
7% error
a1 a2 a3 bl b4 cl
9301.3 9253.1 9303.9 9401.6 9179.1 9627.6
-1.32 -1.83 -1.29 -0.25 -2.61 2.14
a1 a2 a3 bl b4 cl
9284.3 9229.6 9489.1 9347.2 9179.1 9671.0
a1 a2 a3 bl b4 cl
9272.2 9234.9 9588.4 9306.5 9179.1 9693.1
case
a1 a2 a3 bl b4 cl
9255.7 9249.5 9704.7 9257.7 9179.1 9737.6
a1 a2 a3 bl b4 cl
9232.4 9281.4 9845.6 9212.1 9179.1 9831.2
h , = hmax; See Table I for the
E values.
approximation for 2 w k
0.0
a1 a2 a3 bl b4 cl
4840.8 4931.5 5033.8 487 1.1 4755.9 5096.0
- 3.54 -1.74 0.30 - 2.94 - 5.24 1.54
a1 a2 a3 bl b4 cl
4825.0 4879.9 5376.1 4843.0 4755.9 5311.7
-3.86 -2.76 7.12 -3.50 -5.24 5.86
a1 a2 a3 bl b4 cl
4814.2 4892.1 5553.2 4821.9 4755.9 5472.6
-4.07 -2.52 10.65 -3.92 -5.24 9.04
a1 a2 a3 bl b4 cl
4800.3 4923.9 5757.8 4796.6 4755.9 5669.1
-4.35 - 1.89 14.73 -4.42 -5.24 12.96
a1 a2 a3 bl b4 cl
4783.4 4989.5 6003.6 4772.9 4755.9 5932.2
-4.69 - 0.58 19.62 - 4.90 -5.24 18.20
Ca
- 1.50 -2.08 0.67 -0.83 - 2.61 2.60
0.5
A
-1.63 -2.02 1.73 -1.26 -2.61 2.84
1.0
- 1.80 - 1.87 2.96 -1.78 -2.61 3.31
2.0
- 2.05 - 1.53 4.46 -2.26 -2.61 4.30
5.0
h, =
B
C A B
C
A B
C
A B C
&,,.E;
where = x/x,,
Aa
0.173a
Comparison. To facilitate comparison of errors obtained by the various approximation parameters for a given frequency distribution, we have rewritten eq 27:
= h/h"
t;
Ba
t ( b l ) , E(b2), and t ( b 3 ) yield same approximation. See Table I1 for the t; values. h , = hmax/q; [t;(cl), q ( c l ) ] and [t(c2), q ( c 2 ) ] yield the same approximation. See Table I11 for the ( t ; , q ) values. Smallest possible positive k value (= 0.172663) for case C. Frequencies (cm-I): 1648.9, 3833.2, 3943.4.
p
case
(43)
and @'s are proportional to b's as seen in Table IV. The square root of S / ,A, is proportional to the relative error in the ZPE for a model system for which the eigenvalues are continuously distributed according to eq 24:
cm'
I
% error
See footnotes a to d of Table V. Frequencies (cm-I): 666.1, 666.1, 1335.6, 2350.9. a
This measure of relative error has been plotted in Figure 1. Every best parameter, except one, that we have obtained is represented in the figure. For the case of Xo = A-, the optimum parameters, [(al), [(a2),[(a3) are shown as al, a2,and a3,respectively. For the cause of [ = &/X,, the three roots of G2([) = 0 and the corresponding X,values yield the identical curve marked bl, and the root, [(b4) = 4 - 2(2)'12, and the corresponding Xo lead to the plot identified as bq. The other root of Gl(x) = O,[(b5), gives consistently poor results, the error being out of the range of Figure 1. The plot denoted as c1 represents the identical results produced by the smaller pair of roots in the general case. The general trend seen in Figure 1, i.e., that the relative error plotted decreases with increasing k should not deceive one into concluding that one should use as large a k vlaue as possible, because we actually do not have a complete freedom of choice of k: k reflects distribution of eigenvalues of a given molecule. Thus, the best choice of k, i.e., the value which yields the best approximation for a given molecule, would be closely related to the k value that best represents the actual frequency distribution of the given
1012
The Journal of Physical Chemistry, Vol. 87, No. 6, 1983
TABLE VII: Approximations for the ZPE of Formaldehyde, H,12C'60[ Z w i = 11352.3 cm-'1
Oi and Ishida
TABLE VIII: Approximations for the ZPE of Methane, W H , [ Z w i = 19826.6 cm-l]
approximation for 2 w i k
0.0
case
e:
cm-
5% error
Aa
a1 a2 a3 bl b4 cl
11017.7
10 772.2 10 890.0 10 999.8 10 739.5 11 186.2
- 2.95 -5.11 -4.07 -3.11 -5.40 - 1.46
a1 a2 a3 bl b4 cl
10 965.3 10 720.9 11 340.1 10 936.2 10 739.5 11 391.8
a1 a2 a3 bl b4 cl
B' 0.173'
C'
0.5
A B C
1.0
A
B C
2.0
A
B C
5.0
A
B C
approximation for Z, w i
k 0.0
case
e:
cm-'
% error
AQ
a1 a2 a3 bl b4 cl
19 220.0 1 9 104.3 1 9 292.1 19 341.4 18 883.8 19 841.6
- 3.06 - 3.64 -2.70 - 2.45 -4.76 0.08
a1 a2 a3 bl b4 cl
19 163.3 19 016.1 1 9 966.0 19 229.5 1 8 883.8 20 146.6
-3.35 -4.09 0.70 -3.01 -4.76 1.61
a1 a2 a3 bl b4 cl
1 9 123.2 19 036.3 20 325.5 1 9 145.8 1 8 883.8 20 371.7
-3.55 - 3.99 2.52 -3.43 -4.76 2.75
a1 a2 a3 bl b4 cl
19 068.5 1 9 091.0 20 745.2 19 045.5 18 883.8 20 678.7
- 3.82 -3.71 4.63 - 3.94 -4.76 4.30
a1 a2 a3 bl b4 cl
1 8 992.5 19 209.4 21 253.0 1 8 951.5 1 8 883.8 21 142.0
-4.21 -3.11 7.19 -4.41 -4.76 6.63
Ba 0.173Q
CQ
- 3.41 -5.56 -0.11 -3.67 - 5.40 0.35
0.5
A
1 0 927.7 10 732.2 11 586.5 10 888.5 10 739.5 11564.2
- 3.74 - 5.46 2.06 -4.09 - 5.40 1.87
1.0
a1 a2 a3 bl b4 cl
10 875.6 10 764.2 11876.6 1 0 831.5 10 739.5 11 798.6
- 4.20 -5.18 4.62 - 4.59 -5.40 3.93
2.0
a1 a2 a3 bl b4 cl
10 800.4 1 0 837.2 12 229.6 10 778.1 1 0 739.5 12 141.6
- 4.86
5.0
-4.54 7.73 -5.06 - 5.40 6.95
a See footnotes a to d of Table V. Frequencies (cm-'): 1171.6, 1268.3, 1498.7, 1771.7, 2767.3, 2874.6.
molecule. The fact that, in all cases, the relative error generally tends to zero as k goes to infinity is a simple consequence of the fact that, if a molecule has only one eigenvalue, the ZPE can be expressed exactly and analytically in terms of its geometric and force constant parameters. Figure l is, therefore, useful in comparing various approximations at a given value of k rather than as showing the trends with changing k. For the k values greater than unity, the general case gives the best approximation for the model system while, generally yield the better for k < 1, the set ([(all, X,= A-) results than the others. If we explored other roots of F' = 0, they would have yielded numerically even better results,but we chose not to primarily because it would force us to consider an unrealtistically high value for the range of expansion and an unrealistically low value for the normalization reference point, thus making us lose contact with the reality. To be sure, the solutions of the two special cases we considered are also derivable from the general equations, eq 33 and 36, by imposing the respective constraint on them: It can be shown that eq 28 and 29 and eq 30,31, and 32 are derivable by, respectively, setting A. = A,, and C; = Amax/AO in eq 33 and 36.
Examples of Numerical Application The approximation methods developed in the preceding section were applied to the ZPE calculations for water, carbon dixodie, formaldehyde, and methane to test for precision. Since this is solely for the purpose of testing a mathematical formula, the error of approximation for the sum of the normal frequencies was computed by comparing Cui with C(bo+ b,Ai b2X?), both having been computed by using a set of calculated frequencies. The
+
B C
A
B C
A
B C
A
B C
a See footnotes a to d of Table V. Frequencies (cm-I): 1357.4, 1357.4, 1357.4, 1574.2, 1574.2, 3143.7, 3154.1, 3154.1, 3154.1.
F matrix and the geometric pyameters used were for water, those of Papousek and PlivaZ1[R(OH) = 0.9572 A, HOH angle = 104' 30'1; for carbon dioxide, those of Herzberg22[R(CO) = 1.1615 A]; for formaldehyde, those of Spindel, Stern, and MonseB [R(CO) = 1.216 A,R(CH) = 1.09 A,HCH angle = 123.4', and HCO angle = 118.3'1; and for methane, those of Jones and McDoweP [R(C-H) = 1.094 A]. Results of the approximations have been summarized in Tables V-VIII. An inspection of these tables reveals many interesting points. The most striking difference between the results of these tables and Figure 1 is the trend of errors as a function of k: While Figure 1 apparently tells that the relative error decreases with increasing k, it is not the case in the real molecules. But, then, Figure 1is for the model distribution according to which the eigenvalue population, or its importance, changes as Ah, and we have cautioned against the use of Figure 1as showing a trend as a function of k. Some definite trends are noted in Tables V-VIII. (1) Among the case A approximations, [(a31 causes the most k-dependent approximation A, which is in agreement with Figure 1. The strong k dependence makes C;(a3)risky to use in a molecule of unknown frequency distribution. (2) C;(a2)is moderately dependent on k , and it seems to im(21) Papousek, D.; Plha, J. Collect. Czech. Chem. Commun. 1964,29, 1973. (22) Herzberg, G. "Molecular Spectra and Molecular Structure 11. Infrared and Raman Spectra of Polyatomic Molecules";Van Nostrand: Princeton, NJ, 1945. (23) Spindel, W.; Stern, M. J.; Monse, E. U. J. Chem. Phys. 1970,52, 2022. (24) Jones, L. H.; McDowell, R. S. J . Mol. Spectrosc. 1969, 3, 632.
The Journal of Physical Chemistty, Vol. 87, No. 6, 1983
Correlation of ZPE with Molecular Forces
1073
TABLE IX: Best Approximations for the Four Molecules
A B C
0 0.173
" Alternative values:
I
H2O
0.0
hmax
h,,/0.867 17 1 hm,/1.443468
-0.76 -0.25 2.14
2.20 -2.94 1.54
- 3.02
-3.11 - 1.46
-1.77 -2.45 0.08
1.746 080 and 19.836 355.
II
co2
4
2.430 501 0.867 171" 2.438 375
m
0.2
~~
I
I
II
0.4
0.6
0.8
1.0
A/AITlClX
Figwe 2. Comparison of eigenvalue distributions of the four molecules with the normalized weight density function: g vertical dashes in the distribution chart represnt a g-fold degeneracy.
prove as k increases. A further investigation into the large k domain showed that the errors for the four molecules all algebraically increase with increasing It: At k = 10,20, and a (see footnote a of Table I), the relative errors for water are -1.26, -1.05, and -0.76%, respectively: those for C 0 2 are +0.41, 1.16, and 2.20%, respectively; those for H2C0 are -4.01, -3.60, and -3.02, respectively; and those for CHI are -2.64, -2.28, and -1.77, respectively. Thus, the absolute errors for all but carbon dioxide decrease with increasing k . Except for GO2,the best k value is inifinity, for which [ = 2.430501 under the constraint of X, = Am=. (3) Although the smallest root, [(al), for case A is out of range for the assured convergence of the series until k reaches about 5, it gives approximations which are comparable with the other roots. The approximation is moderately dependent on k, and the best k value for this
is zero. (4) Among the case B approximations, the roots of G2(5) = 0, i.e., [(bl), 5(b2), and 5(b3), yield a better approximation than f(b4) in all cases, which is in agreement with Figure 1. The larger root of GI([)= 0, 5(b5), gives much larger errors in all cases, and they are included in neither Figure 1 nor the tables. As discussed earlier, [(b4) gives a result which is independent of k once the frequency distribution is fixed. The variation of the b4plot in Figure 1 is due to the change in the frequency distribution. The best k value for the t(b1) approximation is zero. (5) The best k value for the general case is 0.173. The best approximations in each of cases A, B, and C have been summarized in Table IX. As expected, the best choice for the k value depends on the molecule. In the absence of detailed information about the frequency distribution, zero or values close to zero seems to be a safe choice. In a situation in which the high-frequency end of the spectrum is heavily populated, as is the case in large paraffin molecules, it would make sense to give a relatively large weight to the high end of the spectrum by using a large k, especially when one recalls the fact that, for a given relative error in the approximation of the error for the ZPE is more affected by the higher frequencies. In Figure 2 we have plotted the normalized weight density function, ( k + l)(A/Am=)k, as a function of X/Xm, along with the eigenvalue distributions of the four molecules we have examined. In the distribution a g-fold degenerate eigenvalue has been shown as g vertical dashes. It is seen that the spectra of C02 and H2C0 have their lower ends more populated than the high ends. This is consistent with the relatively poor approximation results obtained for these molecules in our present development when only positive values of k were explored. Generally, relative errors of a few percent are satisfactory, especially for second-order approximations, which would account for interactions between only two internal coordinates taken at a time.25 Acknowledgment. We gratefully acknowledge use of computer time at the Computer Center of the State University of New York at Stony Brook. Supplementary Material Available: A detailed table of best values of 5 and 7 as a function of k (4 pages). Ordering information is given on any current masthead page* (25) See Table I in ref 12.
