James S. Slmsl and George E. Ewing Indiana University Bloomington. IN 47401
Experiments in Quantum Chemistry: The Linear Variation Method
For most problems in quantum chemistry the Hamiltonian operator describing the system is easy to write down. The difficulty is that the resulting Schroedinger equation usually cannot be solved analytically. Fortunately, with digital computers, useful approximate energy levels and wavefunctions may be obtained using numerical methods. The wide availability of computers has expanded enormously the usefulness of quantum mechanics for the chemist and even elementary courses in quantum chemistry and molecular spectroscopy often include their use. Unfortunately elaborate computer programs like complex pieces of laboratory apparatus can appear to the student like mysterious black boxes which deliver answers with little meaning or educational value. In this paper we will show how a single simple computer program can he used to study a multitude of problems in quantum chemistry. These problems include doing Hiickel molecular-orbital calculations, determining energy levels and wavefunctions for the anharmonic oscillator, the hindered rotor, and the perturbed particle in a box, calculating the ground state energy of the helium atom, analyzing magnetic resonance spectra, and so on. The technique used in the calculations we will discuss is the linear variation method. A computer program solves the secular equations to provide eigenvalues and eigenfunctions. Experiment in the dictionary entry is, in part, "to demonstrate some known truth" ( I ) . In keeping with this definition, we have labeled our calculations "experiments." Our examples allow for a wide variety of input data choices and the results either demonstrate theorems of quantum chemistry or may be compared directly to laboratory measurements. The computer, like the spectrometer or analytical balance, is therefore just another experimental tool which when used with care can demonstrate chemical principles. Theory The presentation of the linear variation technique appears in most quantum chemistry textbooks (2-7). The problem is to determine approximate solutions t o the Schroedinger equation,
A,bi = Ej,bi
(1)
for which neither the true eigenfunctions, $i, nor true eigenvalues or energies, Ei, are known. In the linear variation method we approximate the true eigenfunction by a trial function which is a linear combination of n linearly independent functions,
The basis functions, f,, must be well behaved and satisfy the boundarv conditions of the ~roblem. The energy associated with the trial function is given by the variational integral,
'Present address: Mont Alto Campus, Pennsylvania State University, Mont Alto, Pennsylvania 17237. %Theprogram is available from the authors upon request. It requires two subroutines,QCPE program numbers 62.4 and 97, which are available from the Quantum Chemistry Program Exchange, Deoartment of Chemistrv. Indiana Universitv. Bloominrton. Indiana 47401. A listing and punched cards may bebbtained fo; $98.50. The fee is waived in hardship eases. 546 / Journal of Chemical Education
where Hjk = J"f7Rfad~
(4)
and s;k = Jf;fkd.i
(5)
Now the variation theorem assures us that for any choice of the cj coefficients the approximate energy will be greater than the true lowest energy value of A, W>E,
We must therefore systematically all the c ; coefficients . adjust . to minimize IY in order to get H.; gwd an nppn,ximatim as possible to thi, true energy. This isaccumplished hy the condition rifi'ldc, = 0 for i = 1.2, . . . . , n and leads T U the set of equations
2 [(Hlh - SjrW)eal = 0
k-1
i = 1,2,.. . ,n
(6)
These equations have solutions of physical significance only if the secular determinant is zero, HII - S H W H l z - S12W ...Hln - SlnW
I :
Hm - S Z IW Hz2 - S22 W
...HZ,,- SznW ...
I
=0
(7)
T h e n roots, Wo, W z ,. . . , W,-I to this secular determinant are approximations to the exact energies of the original Schroedinger equation. From MacDonald's theorem ( 6 )these energies (when arranged in order of increasing values) provide upper bounds to the true energies of eqn. (1)
E n < W o , E , s W,,...,En-,sWn-, (8) By taking enough terms in the expansion of eqn. ( Z ) , the energies Wi can be made to approximate the true energies Ej to a high degree of accuracy-in some cases one part in
,,",
(1Z).
Any one of the roots W; when substituted into eqn. (6) will generate a set of c, coefficients which then specifies the approximate eigenfunction, $i = @i, of eqn. (1). We, therefore, have a recipe for obtaining solutions to the original Schroedinger equation.' We need only select the functions f, and produce the matrix elements Sij and Hij. The computer program in solving eqn. (6) and (7) can then generate the approximate eigenfunctions and eigenvalues. The Computer Technique Our program,which is put together from subroutines available from the Quantum Chemistry Program E~change,~ is run on the CDC 6600 comouter and samole run times are a fraction of a second for 10 term trial twiatiun fun;tions. Since the mtwe pnrgmm 16 i n liORTR.AN I V , i t can t w implemmted easily t m mwat C ~ I P and ~ P onwersity computers. Some Experiments The Anharmonic Oscillator
The harmonic oscillator is an example treated in all elementary quantum chemistry texts (2-7). The Hamiltonian operator far the problem
we can see from Figure 3, and we have no theorem t o indicate the direction of the approximation errors. Further experiments with oetatetraene may be performed to learn what basis functions are most effective in yielding an accurate energy level by analogy with the quartic oscillator problem. Is it possible to block diagonalize the determinant by appropriate ordering of the basis functions? A study of the shift of energy levels as a function of V,is informative. Examination of the eigenfunctions to learn what funetions are mixed in is revealing. The calculations can of course be applied to the other polyenes from ethylene to vitamin A. The literature (13) is filled with applications of FEMO to cyclic conjugated systems whieh also may be treated by linear variation techniques.
Hindered Motions: The Rotational Stark Effect When an electric field is applied t o a sample of molecules whose rotational spectrum is being recorded, shifts and splittings of the lines are observed. The solution to this problem makes use of the linear variation method and is known t o those familiar with microwave soeetroscoov (14). Simole examoles of hindered rotation ~ r o b l e m s mnv fo&d in the lite'rature~.(15.16). o t h e r linear variation exner, ,he-~~~~ iments w h ~ r hexplorr cplitting oiangulnr momentumstates inrlude the Stark effect vf the hvdruyen ntum ( 2 . 6 1Ltgand . field or crydal field problems (17) are closely related t o these examples. ~~
~~
~~
~~~
Figure 3. Semi-empirical estimates of the pi-electron energy levels of octal* traene using the hepelemon molecular (FEMO)orbital mdai. me dashed line energy levels are obtained from first order perturbation theory. The solid line energy levels were obtained using the linear variation treatment. is in poor agreement with the observed r* + a transition a t 33,100 cm-' (13). The particle-in-the-box model can he improved by modifying the potential V t o be (18)
V=
m
everywhere else
which has the effect of causing more electron density in the region of the double bonds than in the single bond regions. A value of V, = 32,000 cm-1 was selected because i t fits a whole series of polyene spectral data (13). This potential is shown in Figure 3. First order perturbation theory gives for the energy levels - nZr2h2 E,.;EU.+E,-+ S$F$:dr (19) 2m12 E:) - (EP illustrated in Figure 3. The calculated transition (E! E i ) = 29,987 cm-1 is a considerable improvement over the unperturbed result. However some of the discrepancy results from the crude determination of energy levels by first order perturbation theary. An accurate determination of the energy levels influenced by the cosinusoidal potential energy of eqn. (18) may be obtained easily by using the linear variation method. The trial function is an expansion into the particle-in-a-box wavefunctions of eqn. (16),with f l = $7, fz = $!, .I,,= The basis functions are orthonormal so S, = 6," and the energy matrix elements are found to he
+
..
+
$On.
The energy level from an expansion of the trial function into n = 10 hasis functions is shown in Figure 3. The transition energy, Ws - W* = 31,573 nn-I, changes only slightly in increasing the number of h i s functions. The difference between this result and the observed transition a t 33,100 cm-' is therefore due to the FEMO model we have chosen rather than calculation errors. Figure 3 provides a comparison of perturbation theory and the linear variatian method. First order perturbation theory has yielded the approximate energies readily since only eqn. (19) was needed in the evaluation. The linear variation method required the evaluation of n nnrmher of matrix elements as well as solvine " the secular emation. However the variation method eives us accurate results. and Mae~
~
The HyNeraas Problem The pioneering work on helium was done by Hylleraas (18) in the late 1920's. Because the Hamiltonian (given here in atomic units) contains the l l q z term whieh describes the repulsion of the two electrons, 1-1+ 1 , A = -'/2v: -'/*V$ - (21) rI r z rlz The Schroediger equation has not been solved analytically. Hylleraas used variational functions that contained the interelectronic distance r12 explicitly. They show a dependence on the correlation between the motion of the two electrons and go beyond Hartree-Fock treatments. The six term Hylleraas wave function (20)
+ = e-En e-frz
+
l e ~ c
h
+ rZ) + ~ 3 1 +~ c4(c12 2 + rz2)
+ c5r,r2 + c&l
[ = 1.82
(22)
can be used also to provide a striking example of the power of the linear variation method. One can do successive 1.2.. . .6 term calculatima of the energy of the helium ground state nrcm." from whwh one can w e dearly the wnwrgence toward tnr lahorators experi. mental result and obtain an enrryy whieh 19 superiur tcr all but the most rprent cmfiguration interacrion rnlculntloni employing expnnsims of lengths gredrer than 100 basis f m r t i m i . In addition, the ml~.ulalimsdwnonstratr thr rffect g n f elertrm wrrelatiun, which is to make the wave function depend explicitly on the distance between the electrons, in such a way as t o decrease the value of the wave function when the electrons are close together and to increase its value' when the electrons are far from each other.
Hiickel Molecular Orbital Calculations Hiickel Molecular Orbital (HMO) calculations are probably the most popular quantum chemistry exercise. The subject is discussed even in introductory chemistry texts and in many monographs (20). Special computer programs have been offered in this Journal (21-22), yet the same linear variation program we have presented will do the job nicely. As an illustration let us consider the HMO calculations of pyridine. The Hamiltonian (6) in the HMO method deals only with the r electrons which are n, in number and is written as the sum
Since the r electrons are treated independently, the Hiickel MO's, +i, satisfy
~~
= ei+i
A%)+;
(24)
and the energy levels are either above or below the "exact" values, as
where ei gives the energy of each level. The MO's are written as a sum of 2p, atomic orbitals of the n interacting atoms
"atria elements and a table of results can he obtained from the authors upon request.
(25) Z ci,fr r=1 where f, is the atomic orbital on the Phatom. The matrix elements are written down in the usual way. The Coulomb integral is, +i
548 1 Journal of Chemical Education
=
"
H$ = SfXi)@!;fr(i)dui = n c if r is a carbon atom = a N = (3/2)0cc n c if r is a nitrogen atom.
+
HMO Elgenvalues and Elgenfunctions of Pyrldlne
(26)
The resonance or hond integral is H;Ef = Sl:(i)fi:ffs(i)dvi = Pcc for C, and C, bonded = BCN = (4/5)0cc for C, and N, bonded = 0 otherwise.
(27)
=
ea=a+P e, = a
The Zp, orbitals are taken to he orthonormal,
S, = Sf:(i)f&)dui
+ 2.30 e, = n + 1.440 e, = n
6,
(28)
The Coulomb integral for the electron associated with carbon is n c hut is increased ( 6 ) (somewhat arbitrarily), to (312)acc ac t o account for the increased electronegativity of the nitrogen atom. The resonance inteeral for the C-C bond is also different for that of the C-N hond. he small orohlem that results hecause our matrix ele-
+
- 0.430
e,=a-O eg = a
- 1.810
+ + + + + + + +
g, = 0.22t, + 0.26f2 0 3 7 t
074fr + 0.37fr 0.261, g* = o.sof, 0.43f2 o.02fs 0.51t + 0.02fs 0.4316 g3 = - 0.50f20.50fa 0.50fr 0.501, 4, = 0.55f, - 0.12f2- 0.50f3+ 0.411, - 0.50fs - 0.121, gs = 0.50fz 0.5Ms 0.50fs - 0.50f, = 0.531, - 0.48f2 0.341%0.16fr 0.34fr - 0.48fa
+
me
+
+
+
determinant eqn. (7) becomes
The eigenvalues and eigenfunctions of this HMO calculation of pyridine are given in the Table. The eigenvalues are correlated to those of benzene in Figure 4. Pyridine is more strongly bound by its n electrons than benzene essentially because nitrogen is more electronegative than carhon. The double degeneracy of the el, benzene levels is also lifted. The assignment of the symmetry species of pyridine fallows directly from the form of the eigenfunctions listed in the table. These molecular orbitals are sketched in Figure 5. The diameter of the circle is proportional t o the cj, coefficients. The shaded circles indicate that the sign of the 2p, lobe above the plane of the paper is negative while the open circles indicate this lobe is positive. Since pyridine belongs t o the Czugroup all representations, (al, a ~bl, and bp) are nondegenerate. Since either reflection of 41in the plane of the molecule or rotation by 180" about an axis containing Nand C1 changes itssign, this molecular orbital must belong t o bl according t o the character table (6). The other mi's are classified readily. An interesting experiment is to show that if $I,42, 44, gs (belonging t o hi) and h 9 5 (belonging to an) are used as basis functions rather than 11,. .fs the 6 X 6 determinant becomes block diagonal. A 4 X 4 containing only the bl MO'sand a 2 X 2 containinithe az MO'sresults. This illustrates one of the basic theorems of group theory which states that Hjj = 0 and Sv = 0 if gi and gj belong to different symmetry species (3, 6 1 7 ) . Further examoles of MO calculations are obvious. Pi electron
Figure 4. Conelation diagram for benzene and pyridine from Huckel MO calculations.
.
writing down the Hjj and S, elements.
Analysis of Magnetic Resonance Spectrum The linear variation method is the standard technique for calculatine nmr and esr soectra. Comnuter nroerams and aleebraic devices
have already outlined. For our illustration, let us consider the proton magnetic resonance spectrum of 2-bromo-5-ehlorothiophene(24),
This molecule has two protons which experience different magnetic fields and are assiened chemical shifts.. aa.. and m. The interactions of the nuclear spins of these protons is specified t k u g h the coupling constant, J. The spin Hamiltonian is given by ~~
~
a=-",(I - TA)I,A- v,(l - un)l,n+ J ~.is A (30) where 7.4 and is are the operators for the total nuclear spin angular
h(b1) #35(a2) +&I) Figure 5. Symmetry of the Huckel molecular orbitals of pyridine.
momentum of protons A and B and I,n and 1,s are the operators for the components of spin angular momentum along the axis of the applied magnetic field. Using a fixed frequency u, = 30.5 MHz of the resonance spectrometer, the observed spectrum can be explained using la* - an1 = 1.54 X 10-7 and I JI = 3.9 Hz as we shall see. The basis functions used in the expansion of eqn. (2) are formed from the proton spin functions, whereh = (aru) indicates a spins for A and B, fz = l a 0 ) indicates a spin for A and Bspin for B and so on: In= lorn), fd = 100). These functions are orthonormal, so for eqn. (5) we have S,, = 6,? In previous examples the expansion of rp was in terms of a long series of basis functions truncated to keep the secular determinant to a reasonable size. Here the four bases functions form a complete set and the accuracy of the energy values that result from solving eqn. (6)and (7) is limited only by computer errors-usually less than 1 part in 10 (12). In all magnetic resonance problems the complete set of basis functions is usually fairly small. Volume 56. Number 8, August 1979 I 549
Conclusion
The representative experiments that we have described above should serve to illustrate the versatility o f the linear variation method and the ease with which it can be applied using a single program package. We feel that these experiments, and others like them, can be an important l e a r n i n g experience in the study of quantum chemistry. Acknowledgments
The authors would like to thank Mr. William Heffner and Mr. Gene Lamm for assisting us in our use of the computer as a teaching device. We would also like to acknowledee the assistance of Ms. Mary Seaver in the preparation of this ~~
~~~
paper. Literature Cited
of 2-brom*S-chlorothlo-
Figure 6. The proton magnetic resonance spectrum phene.
The energy matrix elements o f eqn. (4) from the Hamiltonian o f eqn. (30) are obtained by the usual spin operator techniques (6.24, 2.5 )
.
+
H 1 1 = ~ ~ ( - %UA+ 1 Hz2 = -%wo(a~
+ '/4J
'/~uB)
- a d - 'hJ
Hz3 =H32 =
'Id
Ha3 = '/ZYO(UA - UB) H44
= uo(1-
- l/&J
(31)
'IWA - %ud + '/4J
(11 ''Webster's New Collegiate Dictionary,). Merriam Co.. Springtiold, 1949. toQumttm Mechhhii."Mfficcaw-Hill, (21 Paullw,L.and W h m . Jr., E. BBBB'Intmdumdtiii NeaYurk, 1935. (31 Eyring, H., Walter. J., and Kimball. G.E.,"Quantum Chemistry," Wiley, NewYork, 19dd .....
(4) I51 (6) (71 (8) (91
W.. Kauzmann, "Quantum Chemistry." Academic Press. New York, 1951. Karplua. M.. and Porter. R.N., "Atoms and Molecules." Benjamin. New Yark, 1970. Lcvine. 1. N.,"Quantum Chemistry." Allyn and Bacon, Boston, 1970. Pilar. P.. "Elementary Quantum Chemistry." McGraw-Hili, New York, 1910. Chan.S. I.,andStelman,D., J. Mo!.Sprr. 10,Zifi (19631. Wilson, E. B., Decius. J. C., and Cross. P. C.,"Muleeular Vibralions." McGrav~HilI, New Ymk. 1955. 110) Somoriai, R. L., and Hornig, D.F., J. Chem. Phyr. 36,1980(19621. (111 Heilbrunner, E.. Rutishauser, H., and Cerson, F.. He!" Chsm. Acfo. 42, 2285
,.""",.
,,O~O>
Since t h e c h e m i c a l a h i f t s n r e mcawrrd r e l a t i v e t o s o m e n t a n d a r d w r r a n ~ u r r e i s l u l l ydiscuss l l w splitting p a t t ~ r nhy a r l , i l r a d y s e t t i n g ~ m r o f t h p m . .n~a .~e a u a l r o a m , . W e t h r w f i m h s v e o , , = 1.51 X NlThe eigenvalu& (;n Hz) and eigenvectors which result are
(121 Tipping. R. H.. J. Mol. Sperf. 59,8(1976). (131 Bayliss, N. S.. Quart. Reu. 6, 319 (1952). Plalf, J. R.. in "Encyclopedia dPhysicsHandbmhder Physik."pdited by S. Flugge, Vol. 37,psrt 2, p. 173. Springer Verlag, Berlin, 1961. MeGlynn. S. P.. Vanquickenborne. L. G.. Kinoshita. M., and Cerroll. D. G.."lntduction to Applled Quantum ChemirUy." Holt.Rinehartand Winston, New York. 1972. (141 Tumes, C. H., and Sehawlow, A. L., "Microwave Spectmscopy," McCraw-Hill, New York. 1955. (151 Hendenon,G..andEwing,G.,J. Chsm.Phys. 59,2280 119731. (161 Lewis, J., Melloy,T.,Cheo,T.,andLaanc,J., J. Mo!.Strucf. 12,427 (19721. (171 Cotton. F. A,, "Chemical Appliestlonr of Group Theow" Intemience, Now York.
The eigenvalues given the energy levels as shown in Figure 6. The spectrum may he calculated f r o m the eigenvectors using the usual relationship ( 2 4 , 2 5 1
1181 See Hyller-. E. W.. "Remini~ceneesimm Early Quantum Mechanics ofTwa E M r o n Atoms,"Rcu. Mod. P h y s 35.121 (19631. (19) See, for example, SlaUr, J. C., "Quantum Theory oiAtomieStrudure,). MeGrav-Hill,
.
,ON, ."",,.
N-w Verb 1960 \in,7
-
I ( ~ ~ I X A + I X B ~ ~ , ) ~ ~
which is proportional to the intensity of the ti i) transition. The calculated stick f i g u r e spectrum is superimposed in the observed spectrum in Figure 6. The agreement is excellent. Other magnetic resonance spectra can be analyzed by analogous methods. While other mathematical methods may generate spectra more r a p i d l y , our simple program does the jab directly, and in a manner we f e e l that it i s e a s y to understand.
550 1 Journal of Chemical Education
Rewmanee: McCraw-Hill. New Ynrk, 1959. (25) Carrington. A.. and McLachlan, A. D., "lntruduetion to Mawetie Rexmanee,"Harper andRuw, New York. 1967.
