ELEMENTS of the QUANTUM THEORY* SAUL DUSHMAN Research Laboratory, General Electric Co., Schenectady, New York
CHAPTER X . THE HYDROGEN HOLECULE. PART 11. THE VARIATIONAL INETHOD
w
Because of the equivalence of the two electrons, it is evident that
METHOD OF SECULAR EQUATION
HILE the HL method yields only an approximate for the energy of formation . . of Hz fromvalue the atoms, more accurate results have been obtained in the solution of this problem as in that of the helium atom, by the application of the variational method. As a first illustration of this, it is interesting to observe how much more readily the eigenvalues may be derived in this manner as with the rather tedious calculation by HL in which the perturbation method was used. As shown in Chapter IX, Part 11, we can write the condition for the solution of the S. equation in the form
SL
HI, = Hgn and HI, = HI,
Hence, equation (450) becomes (aa
Differentiatingwith respect to a and 6, respectively, we derive two relations which must be satisfied. These are:
=
(a'
+ b' + 2abS2)bbE - = 2a(HtI - E ) + 2b(Hil - S2E) = 0 , a
(af
+ ba + 2ab.P) bE bb = 2a(H1. - SZE) + 2b(H,, - E ) = 0 . -
Consequently, we derive the determinantal relation
+H&
E
+ b2 + 2abSz)E = (a2 + bZ)H,, + 2ebHt.
= minimum
1%;
(361)
Jb'd.
13E,2.:
=
0.
that is, the secular equation, where H is defined by equation (412). Let us assume, as in the f i s t part of this chapter . ( H u - E P - ( H I ,- SaE)' = 0 . that the zero-order eigenfunction, $O, is a linear comThe roots of this equation are, evidently, bination of the functions 9 1 = uA(l)ua(2)
(401)
Es =
and 9%= u d 2 ) u d l )
(403)
and E* =
where u r ( l ) , etc., are of the form shown in equation (402). Let +a = a+$
+5 6
Hu
+ HI.
H,,
- H,,
;,+-$1
(419)
HI, = J91~(2& =
+ b2 + 2abP)E = asHl1+ ab(Ht1 + Hm) + b2H2~(450)
2E.
+ VO -
- e3$(&
I
=
Ju+ldr
Hn = = J#fl@dI,
I
(451)
:,
fin - V @ ) ~ Y , ~ V I
1
where
1
(454)
+ + e~(l/rL2)+Lqduldu9
(a2
Hu = s + x ~ 6 d T
(453)
Utilizing equations (412) and (426), given in Part I,
Since and & are orthonormalfied functions, it follows from (448) that
Hit = J + t ~ + l d r
(452)
2Ea
+ G)+Ah~d~)
+ ii +- 31- 2 3 ~
(456)
where JI and Jz are expressions defined by equations (441) and (442), respectively. Comparing with equation (444), it is seen that Hu = 2Eo
T h i s is the fourteenth of a series of articles presenting a more detailed and extended treatment of the suhject matter covered in Dr. Dushman's contribution to the symposium on Modernizing the Course in General Chemistry conducted by the Division of Chemical Education at the eighty-eighth meeting of the American Chemical Society, Cleveland, Ohio, September 12, 1934. The author reserves the right to publication in book form.
330
+ En
In the same manner it is H,S =
J + , h p ~+. vo -
=
s3(2&f $) + eP J
(466)
that VZ* - K ~ J ~ V , ~ V ~ ( I / I ~ ~ ) Q ~ ~ ~ V ~ ~ Z ~
(457)
each atom contains only one electron which is effective for interaction. I n the molecular orbital method, we represent the eigenfunction for the molecule by
where Kzand Kl are the expressions defined by equation (445) and (446), respectively. Comparing with equation (447), it follows that
where a capital P ' is used, as Van Vleck and Sherman have suggested, to denote the wave function for the Hlr = 2EoSa ELP (458) whole system, and small II. for the function of each electron moving in the field of the two nuclei and the Hence other electron. + E1r = 7.. [see equation (432)] Es - 2Eo = En ----Let +A denote the wave function for the motion of 1 +S1 one electron in the field of the nucleus A (so that 3, - 231 K I - 2SK. e' =fi+ (459) +A is an atomic orbital), and similarly for +B. l + 9 Then equation (461) becomes, while
+
+
- En EA - 2Eo = En -=
- S' q~ [see equation (435)l =caz +A - 232I --KSP, + 2SK2 1
(460)
It will be noted that a term such as +A(l)+a(2) implies that both electrons are associated with nucleus A, while +B(1)+~(2)means that both electrons are associated with B. Therefore, the right-hand side METHOD OF MOLECULAR ORBITALS of (462) represents a condition which may be deIn the HL method the eigenfunctions for the Hz scribed as follows: (1) The system has a probability of being in molecule are obtained a s linear combinations of functions which are products of atomic orbitals, that is, either or both (depending upon the ratio dB)of the one-electron wave functions which describe the motion ionic states A- : B+, or A+ : B-. of an electron in the field of only one atom. On the (2) There is a probability, defined by azp, that other hand, F. Hund, R. S. Mulliken,' and others have the molecule will exist in the homopolar state A : B, investigated the possibilities of developing wave in which the bond is non-fiolar. The HL argument takes into account only the second functions which shall express the motion of each electron in the potential field resulting from all the possibility and neglects completely the possible ocnuclei and other electrons present in the molecule. currence of ionic states. ' Obviously the actual wave function should, in Thus, in this method the system Hz is regarded as the result of the addition of an electron to the ion Hz+. general, be represented by a combination of ionic and The point of view is therefore analogous to that of homopolar functions, and for purposes of further discussion, we may write equation (462) in the more Hartree in the case of atomic systems. A comparison of the results obtained by the two compact form, methods has been made by Van Vjeck and Sherman = a+< &EL (463) and their conclusion is as follow^:^ where $i designates the wave function for the ionic "It is hard to say categorically whether the method of molecstates, #EL that for the homopolar state (HL form of ular orbitals or the HL method is the better. The latter unfunction), and it is necessary that doubtedly is much preferable at very large distances of separation These results are identical 'with those derived in equations (443) and (447).
*
of the atoms, for then the continual transfer of electronic charge from one atom to another demanded by the ionic terms surely scarcely occurs at all. On the other hand, at small distances, the HL method probably represents excessive fear of the ra effect, and the factorization into n one-electron problems presupposed by the method of molecular orbitals may be quite a good approaimation."
The reference to "ionic terms" requires further amplification, and, in order to illustrate the significance of this point, we shall consider the application of the two methods to a diatomic molecule AB in which R. S. M~LLIKEN, Rm. Mod. Phys., 4 , 1 (1932). See also J. H. VANVLECKAND A. &=AN, ibid., 7, 168 (1935) for a comprehensive discussion of this topic and bibliography. a VANVLECKAND SAERMAN, 1 0 ~ cit., . p. 171.
+
From this point of view, the distinction between ionic and homopolar compounds is not nearly so sharp as we ordinarily assume. As a in equation (463) varies from 1 to 0, we pass from the completely ionic to completely homopolar type of molecule. In this connection the comments of Van Vleck and Sherman are very pertinent. "There are elements of truth," tbey write, "in the old-fashioned chemistry that HCI has the structure H+Cl-, as the true wave function of HC1 is expressible as a linear combination of various idealized types, and certainly H+CI-must be given some representation.. . .One great service of quantum mechanics is to
show very explicitly that all gradations of polarity are possible. so that in a certain sense it is meaningless to talk of such idealiaations as homopolar bond, heteropolar bond, covalent bond, dative bond, etc."%
The reader will find further interesting remarks on this topic in Pauling's paper,4 "The transition from one extreme bond type to another," in which he uses as illustrations of such transitions from ionic to homopolar molecules the alkali and hydrogen halides. THE DISSOCIATION ENERGY OF H2
These considerations have been found to be important in the calculation of the energy of formation of Ha from the atoms. (This will be designated by D ) For a symmetrical molecule, such as Hz, we usually regard the value of a in equation (463) as insignificant. However, S. Weinbaum5 has shown that i t is possible, by using a function of the form given in equation (463), to obtain a more accurate -value of D,. He assumed
when the two atoms are brought together (the polarization effect). "The simplest way," he writes, "to represent this distortion is to consider the radius of the atom to change with the distance from the other atom. This is effectively what Wang did in his calculations, and it led to a definite improvement in the energy value. However, since the perturbations involved are not spherically symmetrical this cannot be a very gwd approaimatian to the true state of affairs,and the next improvement that suggests itself is to introduce a change in the wave function that will depend on the direction with respect to the molecular axis and will be greatest in the direction of the latter. Since the interactions can be thought of roughly as being along this axis, it seems likely that the electron cloud tends to bulge out in the direction of the second atom."
I n accordance with this idea, Rosen used a modification of Wang's "trial function," as defined in equation (465) of the form,
where OAI is the angle .between TAI and R (the line joining the nuclei), @ is a variable parameter, and C is the normalization constant. Substituting this function in the variation equation (361), the effect was first determined of varying the effective nuclear charge or, assuming p = 0. The value of a! thus and similar expressions for the other hydrogen-like obtained which corresponded to a minimum value of functions. This expression for T was then used E was then substituted in equation (466) and used to in the fundamental relation (361) to calculate a value calculate a minimum value of E for variations in p. The final result led to a value for D,of 4.02 v.e., with d~ ?JE of E, subject to the conditions, - = 0 and - = 0. the corresponding values of the two parameters, a! = bc bZ The lowest value for D,obtained in this manner was 1.19, and p = 0.10 for the equilibrium distance, R = 4.00 v.e., with the values c = 0.256 and the effective 1.416 ao. The most accurate value for D,has been calculated unclear charge Z = 1.193. by H. M. James and A. S, Cb~lidge.~They used a E. A. Hylleraas6 also used the method of molecular separate set of elliptic coordinates (A,, PI and AS, k ) orbitals, starting with wave functions which were for each electron and introduced a new coordinate solutions for ionized hydrogen molecule, HZ+. The p = 2r12/R, thus taking into account a factor which value thus derived for D,was 3.6 v.e. Passing now to the consideration of other varia- had been neglected (because of mathematical diffitional treatments in which atomic orbitals have been culties) by previous investigat~rs. As trial function used, we iind that S. C. Wang7 used modified HL they used the series " Y = ZC[mnjkfi] function of the form where where C i s a normalization constant, and Z, as before, represents an effective nuclear charge which was determined by solving equation (361), subject to the condition ?JE/dZ = 0. A more elaborate expression for T was utilized by N. R o ~ e n . He ~ took into account the distortion in the charge distribution in each atom which must occur
and the summation extends-over positive value of the exponents (including zero), "subject to the restriction required by nuclear symmetry that j k must be even, and taking as many terms as shall prove necessary give an acceptable approximation for the energy." It was fonnd that with R = 1.40~0and 8 = 0.75, five terms in the series were sufficient to give a value for- E much better than any previously obtained. a VAN VLECKAND SHERMAN. loc. cil... . n. 171. The author ~ i$ ~ responsible for the italicized paits. Actually, calculations were carried out with as many ' PAULING, J. Am. Chem. Sac., 54, 988 (1932). See also the as 13 terms in the series. Table 1 taken from their AND J. B. W~SON, JR., "Introduction .discussion by L. PAWLING paper gives a comparison between their results and to quantum mechanics," p. 345. S. W E M B AJ. ~ ,Chcm. Phys.. 1 , 593 (1933). those of previous investigators. ' E.A. HYLLERAAS, 2. Physik, 71. 741 (1931). I S . C. WANG,Phys. Rev., 31, 579 (1928). ' H. M. JAMBSm u A. S. COOLTDOE, I. Chcm. Phys., 1, 825 a N . ROSEN, ibid., 38, 2099 (1931). (1933). ~
+
TABLE 1
~ ~ ~ c i i o ~
oneterm 6 terms 11 terms 13 terms W~thoutn r ~eitler-~ondan
sugiura Waog RO="
observed
D.
(c~ccnan-ao~rs)
R/M
1.40 1.40 1.40 1.40 1.40 1.40 1.51 1.42 1.416
2.56 4.504
4.685 4.698 4.27 2.9 3.2 3.76 4.02 4.72
1.40
It is of interest to compare these calculated values of D. with the results of observations on the energy of
dissociation of Hz, which we shall designate by Do. Direct thermochemical determination by F. R. Bichowsky and L. C. Copeland10 gave the value Do = 4.55 + 0.15 v.e., while H. Beutler" has deduced from the observations on the vibrational energy levels of Hz in the normal state, the value Do = 4.45 (= 102,480 cal.). Since the energy minimum (-D,) as calculated by the methods of quantum mechanics includes the zero-point energy ('/zhvo = 0.27v.e.,asshownin the following section), the observed value of D, is 4.45 0.27 =4.72 v.e., which is in extremely satisfactory agreement with that deduced by James and Coolidge.
+
PART IZI. VIBRATIONAL AND ROTATIONAL STATES OF TBE HYDROGEN MOLECULE* GENERALREMARKS between the values U(r) = -D and U(r) = 0 . Figure ~h~ existence of band spedra, and the observations 57 shows such a series of vibration2 energy states on the variation in specific heats of gases with temper- for the molecule Hz in the normal (1s12,+)state, and ature, lead to the conclusion that, in addition to the the potential energy curve derived in accordance with energy of excitation of electronic levels, the molecules Morse's equation, as explained in the following section. also possess both vibrational and rotational energy due to the motions of the nuclei. Since the frequencies of these motions are small compared to those of the electrons, the electronic motion can adjust itself relatively instantaneously to the motion of the nuclei as if the latter were centers of force a t rest. It is therefore possible to consider the energy states arising from nuclear motions as superimposed-upon the electronic states. Let us consider a diatomic molecule, such as Hz. As shown in Part I, the potential energy for any two atoms, A and B, which combine to form a molecule AB, is a function of the internuclear distance, r, which may he represented graphically by a curve such as that shown in Figure 57. Let U(r) designate this function. By definition the force between the nuclei is given by F(r) = -dU/dr
At the value of r = 70, for which U is a minimum, the force vanishes, while it is negativp to the right of the minimum (corresponding to a net force of attraction) and positive to the left of the minimum (corresponding to a net force of repulsion).t If we assume U(r) = 0 for r = m , then the energy a t r = ro (which is negative) corresponds to the dissociation energy which is usually designated by -D. For values of U(r) > -D, there are two values of r a t which U(r) has the same value. These correspond to mean points of equilibrium for the vibrational motion of the nuclei, and the total energy is given by En = U(r) D. This energy is quantized so that there exist a series of vibrational energy states, designated by the quantum numbers n = 0 , 1, 2, 3, etc.
+
rrmm
, , , , , , , , ,
, , , , , , ,
FIGURE 57.-POTENTIAL ENERGY FUNCTION AND VIBRdENERGY LEVELS FOR NORMAL STATE OF HI MOLE-
TlONAL CULE
It is also observed that with any given vibrational: state there are associated a series of rotational energy states, designated by the quantum numbers K = 0,. 1, 2, etc. Thus, any line in the emission s ~ e c t m m o fa. mo'ecule may be as a transition from an
T h e most recent discussion of the topic of band spectra is a publication of H. SPONER, "Molekulspektren und ihre Anwendune auf chemische Probleme." two volumes.. Tulius Sorineer. Berlin, 1936. lo F. R. BICHOWSKY AND L. C COPELAND, I. Am. C h . Soc.. t See also remarks at the beginning of Chapter VII, J. CHEM. 50, 1315 (1928). EDUC,12, 581 (1935). H. BEUTLER,Z. physikal. Cfwn., BZQ, 315 (1935),
-
. " .
"
upper rotational level K' associated with the upper vibration level v' and with a higher electronic state to a lower rotational level K", which is associated with a vibrational level v" and a lower electronic state. That is, if ;represents the wave-number* of the emitted line, v = Y. + G .' - Go" + F1(u,K) - Fr(v, K) (467)
state lies in the ultra-violet region, as shown by the values of X (in Angstroms) which are attached to the different lines.
-
where ; , denotes the wave-number difference between the lowest levels associated wither' = v" = 0 and K' = K" = 0, G.' corresponds to the vibrational energy, and Ff(v, K) to the rotational energy associated with the hixher electronic state and similarly for G". and Fn(v, K). Thus Figure 58 shows the lower vibrational energy levels for the molecule Hp in the normal (Is%,+) and in the fmt excited (2p1Z.+) states.t The latter state results from the interaction of one electron in the normal (Is) state and another in the 2s-state. The difference in energy for the states v = 0, K = 0 is equal to 90,201 cm.-I (= 11.13 v.e.) and the spectrum emitted by transitions from the higher to the normal
F I G ~ E~~.-ILLuS~~ATING TRANSITIONS
BETWEEN
ROTATIONAL ENERGY LEVELS ASSOCIATEDw m ~TWO DIPRERENT VIBRATIONAL ENERGY LEVELS EOR H, MOLECULE
Figure 59 shows the rotational energy levels associated with the level, v' = 3 , in the upper electronic state and the level, w" = 1, in the normal state of the Hz molecule. The values of ;on the right-hand side are taken from the corresponding values in Figure 58,t while the values under A; give the total increase in rotational energy (in terms of wave-numbers) as the quantum number K is increased from 0 to higher values. The transitions indicated arf: those corresponding to the line L in Figure 58, where K' designates the upper levels and K", the lower levels, respectively. It will be seen that the "head" of the band, corresponding to the transition from K' = 0 to K' = 0, has the value ; = 89896 (or X = 1112 A). This is shown in Figure 58 as the third line from the left.;. VIBRATIONAL ENERGY STATES
For the H2molecule in the normal state, the equilibrium internuclear distance, as shown in the previous sections, is ro = 0.74 A. As the molecule acquires increasing amounts of vibrational energy (due to iucrease in temperature of'the gas), this intemnclear distance increases. Assuming that the restoring force *;=!=' where c = velocity of light. and X = wave- acting on each atom in a diatomic molecule, AB, is length. I t is customary to give energy d8erences in terms of proportional to the displacement from the position of wave-numbers. For the purpose of converting to other units equilibrium for minimum value of U(r) (Hookers of energy, the following relations are useful. (See R. T.BIEGE, law), i t is possible to derive relations for both the Phys. Rev. Supplement. 1, 1 (1929) for values of physical conpotential energy as a function of r a n d the frequency of stants.) vibratiou.12 1electron volt (v.e.) = 8106 cm.? = 23,055 cal./mole of $The difference between the values 4162 and 4157 for = 1.591 X ergs. the level u" = 1 is of no significance. c m . 3 = 2.844 cal./mole R. DE L. KRONIG."The ootical basis of the theorv of vat The values of the enerrnr levels for H. indicated in this and lency," p. 83, published by he Macmillan Company, New th; other figures are taken-&om the extremely interesting paper York, 1935. The reader will find this treatise extremely valuable Phys. Reu.. 44, 165 (1933)on "The emission in connection with the discussion on band spectra and their by C. R. JEPPESON. 7
spectrum of molecular hydrogen in the extreme ultraviolet."
significance in relation to chemical properties.
We shall consider the general case of a diatomic molecule consisting of atoms A and B. Let P A and bB denote the masses of the atoms; let r denote the distance between the two nuclei for any given vibrational state, and let ro designate the value of r for U(r) = -0. Then, the equations of motion for the two nuclei are as follows (see Figure 60) :
where S = S(P)is the "radial" function, K = 0, 1, 2, etc., and a2 = 8 r z p / h 2 . 1 Put S(p) = - 4(p). Then equation (471) becomes, P
.
If we substitute for U(p) from equation (470), and let K = 0 , that is, consider only the case in which there is zero rotational energy, and only vibrational energy, equation (472) becomes*
This equation is evidently similar to the equation for the h e a r harmonic o s c i l l a t ~ r and , ~ ~ has as eigenvalues the series defined by the relation PIOURE~~.-ILLUSTRATIN~ THE VIBRATIONAL MOTIONOR Two NUCLEI
Hence,
"reduced" mass of .molecule. + As shown in Chapter I!V,ls the solution of this equa-
where u = B'A' '
=
PA
!JB
tion is Y
1YO
=
A sin (Zwd
+ 6)
where A is the maximum amplitude, and
The potential energy function corresponding to equation (468) is evidently parabola with vertex a t r = YO, the equation for which is
where v = 0, 1, 2, etc., corresponds to the vibrational quantum number. It follows that the vibrational energy levels should be equally spaced. Actually, the distance between successive energy levels decreases, in the case of homopolar diatomic molecules, with increase in u. That is, the atoms do not behave as simple harmonic oscillators, and the motion is of the anharmonic type, with the result that to a first approximation, the vibrational energy must be represented by an expression of the form where x is a constant for any given molecule. For E = 0, that is, wheq dissociation occurs, it is evident that dE,/dv = Q. Consequently,
that is, (er
+
l/p)
and
Q
U(Y) = -D
k + Z(Y -
I = 2X)
0 '0)'
hw =2-2x
lucwa hw2 4%' 4%
(476a)
or Substituting from equation (469).this can be written in the form where p = r - ro,and vois replaced by wo, in accordance with the notation used in the literature on band spectra. This is the function which must be inserted in the S. equation for the system in order to determine the discrete energy states. This equation was deduced in Chapter VI," and has the form
" J. CXEM. Enuc., 12,381 (1935). 1 '
Equation (210). Chapter VI, ibid., 12, 531 (1935).
D w8c -= hc 4w@Xc1
(476b)
This gives the dissociation energy in terms of wave numbers. Hence, to a first approximation E. = -D
+ hwa(v +
I/$)
3 + '/#077)
- h'
2
(v
As shown by Morse, a potential energy function which is in agreement with this relation and with the
* The solution discussed in the following section is that given by P. M. MORSE, Phys. Rev., 34, 57 (1929); also in CONDON AND MORSE, "Quantum mechanics," Chapter V. l6 J. CHEM.EDUC.. 12, 381 (1935). Chapter IV, equation (lo&).
requirement that E, shall be a minimum for may be chosen of the form
p =
0
where p is obtained as follows. For small values of p, the last equation becomes U(p) = - D
+ B'pPD.
This evidently has a minimum value, U(p) = -D for p = 0, that is, the minimum occurs a t r = 70. Comparing this with the equation (470) it follows that BPD = 2+woa.
and substituting for D, from equation (476a) p =
4-
= 0 . 2 4 5 4 ~ 0 ~
(479)
where
M
= r X 6.064 X 10". = mass in terms of oxygen = 16.
Now C. R. Jeppesonl"ound that for values of the vibrational quantum number from v = 0 to v = 11, the difference in wave-numbers between successive levels is given by the relation,
The values of AG, thus calculated are shown in the column on the right-hand side of Figure 57. FOE o = 1, AG" = 4161.70, and i t will be observed that the value of AG, decreases with increase in v. More recently H. Beutler" has been able to determine, from observations on the lines in the extreme ultraviolet spectrum, values of AG, for v = 12, 13, and 14. If the values of AG, are plotted as ordmates against 2 AG. = *-" "-"
Y,the total increase in wave-number
from u = 0, to v, the resulting curve (see Figure 61) shows that AG. = 0 for '; = 36,100 cm.-'. Therefore, this value must correspond to t.he dissociation energy, Do, of Ht in the normal state. In terms af energy units, ? Do = 36.100/8106 = 4.454 v.e. = 4.454 X 23,055 = 102,680 cal./mole.
From equation (480) i t follows that for o = -%, AGO = 2176.1. This therefore corresponds to the zero-point energy (hvo/2), and must be added to Do to give D, the minimum value of the potential energy, U(p). Thus. D = 36.100 2176 = 38,276 em-' = 108,800 cal./mole. For Hz (M = 1.008), the value of p according to eauation (479) . . is 2.00. Hence the ~otentialenerw function for Hz, as derived from the spectroscopic vibrational energy levels is given, i n terms of wauenumbers, by the relation
+
"
