1549
EXPRESSIONS FOR MULTIPLE PERTURBATION ENERGIES formed will undergo reaction 8, i.e., about 3.6% of the total CH2 will react with P r as VH2 to yield C4H10. Combining the predicted contributions of lCH2 and 3CH2reactions, we calculate R(i-B) = 1.8, in reasonable agreement with experiment without the invocation of 3CH2 insertion reactions. The reaction of lCHz with Nz to recycle CH2 through D M will increase the contribution of ‘CH2 reactions to overall C4H10 yields, while depletion of D M will increase the 3CH2 component. (If information on the depletion of D h l in these experiments was available, the relative sizes of these effects could be assessed and the relative rates of reactions 10a and 10b estimated.) At lower [Pr]/[DM]o, Rabinovitch finds a lower R(i-B),which is consistent with the decreased role of reaction 5 with the higher propor-
tion of scavenger. With [K],: [Pr]: [Nz] = 1 : 6 : 6 X 1110, R(i-B) = 1.55, consistent with the lower reactivity of I< toward 3CH2. The suggestion here, that the large percentage of ‘CH2 reaction products in inertgas-diluted systems is due to relative reactivities and not to large absolute amounts of lCH2,mill presently be tested via absolute quantum yield measurements in DRI-Pr-inert gas systems.
Acknowledgments. I am indebted to the Petroleum Research Fund, Grant 3275-B, administered by the American Chemical Society, for partial support of this research. Funds to purchase the gas chromatographic detector and electrometer were provided by the Simmons College Fund for Research.
Expressions for Multiple Perturbation Energies1 by D.P. Chong Department of Chemistry, University of British Columbia, Vancouver 8, B . C., Canada
(Received September 16, 1970)
Publication costs assisted by the National Research Council of Canada
New expressions for multiple perturbation energies have been derived to complement the old. Possible applications in experimental as well as theoretical chemistry are discussed. In double perturbation theory, the Hamiltonian for the perturbed system can be written as X = Ho
+ AH1 + /a,
(1)
where Ho is an unperturbed Hamiltonian. I n this work, we consider only the case in which (a) the eigenvalue of H o for the state of interest is nondegenerate, and (b) Ha, HI, and Hz are all Hermitian. The eigenfunction and eigenvalue of X are expanded in the usual2 double power series m
m
X1pmjl,m)
= Z=O m
m=O m
(3)
The calculation of E(LIM)requires knowledge of the perturbed wave function up to some Il,m). Great emphasis has been placed in keeping I as low as possible. For example, the well known Dalgarno’s interchange theorem2 gives E(1,‘)= (0,1/HlIO,0) (O,OIH110,1) (4)
+
E(1,Z)= (0,21fJ110,0)
+ (0,1l~lIO,1)+ (o,ol~Ilo,2>( 5 )
I n a recent paperj3 Tuan extended Dalgarno’s interchange theorem to higher orders. Two examples of her eq 12a are
+
E(221) = ( O , O I ~ l ’ l ~ , (0,1IHl’ll,O) ~~ B(2,2)= (0,01H1’/1,2)
+
+ (0,2/H1’11,0)
(0,1l~l’l1,1)
(6)
(7)
where H1’ = H1 - E(lpo). I n this work, we (a) present an expression for E ( L , M ) which requires low ( I m) in ll,m), (b) extend Tuan’s work to triple perturbation, and (c) point out some possible applications of these new expressions for the perturbation energies. Double Perturbation. I n contrast to Tuan’s two model^,^ we consider both perturbations together. Instead of eq 1-3, we write
+
(1) Supported by grants from the National Research Council of Canada. (2) J. 0. Hirschfelder, W. B. Brown, and S. T. Epstein, Advan. Quantum Chem., 1, 255 (1964). (3) D. F.-T.Tuan, J. Chem. Phys., 46, 2435 (1967). Her summation limits are all correct only if one regards all wave functions of negative orders as zero.
T h e Journal of Physical Chemistry, Vol. 76, No. 10, 1971
1550
D. P. CHONG X=Ho+U
(8) (9)
E
=
(lo)
s=o
where
U
=
AH1
+ pH,
(11)
+
E(%’)= (O,l~Hl’~l,O)
(1,OlHl’IO,1)
+ (1,OIHz’ll,O)
where H1’ = H1 and Hz’ = H 2 Equation 20 is just an example of eq IV.12 of Hirschfelder, et aL2 Triple Perturbation. Let us now consider the Hamiltonian X =
Ho
+ AH1 +
+ vH3
(23) where all the operators are Hermitian, and the triple power series expansions m
P E
=
=
m
Z=Om=O
pHz
m
AEpmvnll,m,n)
(24)
ALpMyNE(L,M,N)
(25)
n=O
555
L=O M = O N = O
It is easy to extend Tuan’s eq 12a. The expression requiring low 1 in Il,m,n) becomes
It is well k n ~ w nthat ~,~ =
E ( L , M J= )
(@(-qqp)) -
N
Af
m=O n = O d
(1 - 6~1) k=l
where S > 0; B is S/2 for even S and (S - 1)/2 for odd S; and A = S - B - 1. If we substitute eq 11-13 into eq 14, regroup terms, and match t,he coefficients of ALpM, we get (after several pages of algebra)
c(l,A - llHllL - 1 1, B - L + I + 1) + (1 (A -
E ( L , M= ) (1 -
6~0)
aSl)
(p, A
j=o
h
+ m + 1, M p
+ 1+j
m
max(0,L-Ic-B+j) max (0, L - IC - p )
< p < min (L, A
+ 1+j
(16) (17)
- k)
(18)
< q < min ( R - j , L - p )
(19)
If all the operators are real, a factor (1 - 6 ~ 2 ) can be inserted int,o the last term of eq 15, and simplifies the expressions for E ( L , Mwith ) L M = 2. For example
+
E(I1’) = (0,01H1’10,1)
+
+ (O,OIHz’jl,O)
E(1t2) = {0,1l~l’IO,1)
(1,01H2’10,1)
+ (0,1IH2’/1,0)
T h e Journal of Physical Chemistry, VoL 76,No. 10, 1971
+ 1+
can be inserted into the second term of eq 26, and simpli. example fies the expressions for E ( 2 , M , N )For
+
+
E(liltl)
- - PI% B - j - 4) (15) (L + M ) > 0; A , B, k , a n d j have the
< 1 < min ( A , L - 1) - B - 1) < m < min ( A , M - 1)
q=o
+ m + n) in Il,m,n) is E(191’1=) (0,0,1~H~’~0,1,0} +
lowest (1
q
max (0, L - B - 1)
(C
p=o
- k,p,gld
+
- m - 1) C E P - P - Q ,k - - L + P + C x
i
M-m N - n
- j , M - m - p , N - n - Q)] (26) where d is L/2 for even L and (L - 1)/2 for odd L, and c = L - d - 1. If H1 is real, a factor (1 - 6,2) j
+
(20) (21)
+
(Ojl,OIHi’~O~O~l)(O,O,lIHz’I1,OjO) (1,0,01Hz’10,0,1) (0,1,O/H3’(1,0,0) 4-
I n eq 15, S = same meaning as in eq 14. The new limits of summation are
max (0, M
iz-1
E(lctnztn) C
E(l,l,l)= ( ~ , ~ , o l ~ l l o , ~(0,0,1~H1~0,1,0) ,l) ~ o , ~ , o l ~ l l o , (0,171 o , ~ ~lHllO,~,O) (27) In cont,rast, the formula for which requires the
~MO)
(1 -
{(c,m,nlHlld, M - m, N - n ) -
1
m,mlHzlB - M
(22)
+
(1,0,O/H3’b1,0) (28) H z - E(Otl)O),and
where
H1’= H1-
H3’
H 3 - E(O,O,’).
=
E(18,0Hz’ ), =
Discussion The new expressions in eq 15, 26, and 28 may be useful t o experimental as well as theoretical chemists. If finite basis sets are used and the matrix representations of Hot HI, and Hz can be regarded as known quantities, then the use of eq 15 is relatively simple. A comup to ( L puter program has been written for E(LsM) M ) = 7 and has been submitted to Quantum Chemistry Program Exchange. Consider H = Ho AH1as the true Hamiltonian of a system. Exact first-order properties are given bs2 ( W ) = E(0,1)+ XE(1,’)+ X2E(28’) + , , . (29)
+
+
EXPRESSIONS FOR MULTIPLE PERTURBATION ENERGIES The first-order correction E ( l > lcan ) be easily evaluated by Dalgarno’s interchange theorem expressed in eq 4. To calculate the second-order correction E ( z , l )Tuan’s , expression requires 11,l) in eq 6; our new formula in eq 22 needs only Il,O) and l0,l). If HI involves the usual two-electron operator, then it is still difficult to determine Il,O) exactly. However, we hope that eq 22 will stimulate some interest in the calculation of approximate second-order correction to first-order properties using approximate (l,0).4 For second-order properties such as electric polarizabilities, eq 15 does not lead to simple expressions. One has to use Dalgarno’s interchange theorem expressed in eq 5 for the first-order correction and Tuan’s formula expressed in eq 7 for the second-order correction (if the necessary perturbed wave functions are obtainable). For second-order properties of a second type, such as chemical shifts and nuclear spin-spin coupling constants, eq 27 provides a simple way of calculating the first-order correction (if the necessary functions can be determined). One could, in principle, use eq 26 for higher order corrections; however, the perturbed wave functions required are extremely difficult to obtain. Theoretical chemists interested in physical properties should see that our new expressions have been derived to complement Dalgarno’s interchange theorem and Tuan’s extension, not to replace them.
1551 For some experimental chemists, the new expressions may be helpful in an entirely different manner. Unknown parameters in a Hamiltonian, such as force fieldss and spin are often extracted from spectroscopic data by least-squares analysis. The iterative procedure usually employed has been based on first-order multiple perturbation theory and multidimensional simple Newton-Raphson corrections. With the aid of eq 15 and 28, it becomes quite feasible to use third-order multiple perturbation theory with multidimensional extended Newton-Raphson corrections.* One of the advantages of this new scheme is that it would reduce the chances of going from a safe initial set of parameter values to a set a t which the sum of the squares is a local, but not true, minimum.
(4) D. M . Schrader has indicated an interest in this direction. (5) J. Aldous and I. M. Mills, Spectrochim. Acta, 18, 1073 (1962); 19, 1567 (1963). (6) S. Castellano and A. A. Bothner-By, J . Chem. Phys., 41, 3863 (1964). (7) H. M. Gladney, IBM Research Report RJ-318 (1964); H . M . Gladney and J. D. Swalen, Quantum Chemistry Program Exchange, Program No. 134. (8) J. A. Hebden, “An Electron Paramagnetic Resonance Study of Molecular Spin Multiplets with S 1 1,” Ph.D. Thesis, University of British Columbia, 1970. The fine-structure parameters for the ground-state triplet of pnitrophenyl nitrene were successfully extracted in this manner.
The Journal of Physical Chemistry, Vol. 7 6 , No. 10,1971
