The Ionised Hydrogen Molecule

The Ionised Hydrogen MoleculeAuthor(s): A. H. WilsonSource: Proceedings of the Royal Society of London. Series A, Containing Papers of aMathematical and Physical Character, Vol. 118, No. 780 (Apr. 2, 1928), pp. 635-647Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/94927 .

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lonised Hydrogen Mllolecule. lonised Hydrogen Mllolecule.

These solutions fail altogether near certain points, and it is necessary to use the

expansion with care. The value for , is by no means certain, and it is usually necessary to use the continued fraction (4) ? 7 to check any results obtained from the asymptotic expansion.

Any solution of the equation can be expressed as a linear combination of these two solutions,fl, f2. To obtain the odd and even solutions of ? 7 we must satisfy the following relations,

f(I) =f(r - E) for the even functions, and

f(E) -f( -(-) for the odd functions.

Our approximations do not distinguish between the values for ji corresponding to the even and odd functions, which are

f1i() -t f2(S) an even function, and

fl(.) -f2()) an odd function.

The Ionised Hydrogen Molecule.

By A. H. WILSON, B.A., Emmanuel College, Cambridge.

Communicated by R. H. Fowler, F.R.S.-Received December 19, 1927.)

1. Introduction.

The model which has been proposed for the ion of the hydrogen molecule H2+, consists of one electron and two protons. Since the mass of the electron is negligible compared with that of the protons, we may, to a first approximation, consider the protons as at rest. The system is then a particular case of the problem of three bodies, and can be solved completely classically. This has been done by Pauli,* and more recently by Niessen.t The value obtained by Pauli for the energy of the normal state is not in agreement with the experimental result inferred from the ionisation potential and heat of dissociation of the molecule. Niessen obtains the experimental result by the introduction of half integer quantum numbers.

* 'Ann. d. Physik,' vol. 68, p. 177 (1922), t 'Z. f. Physik,' vol. 43, p. 694 (1927).

2T 2

These solutions fail altogether near certain points, and it is necessary to use the

expansion with care. The value for , is by no means certain, and it is usually necessary to use the continued fraction (4) ? 7 to check any results obtained from the asymptotic expansion.

Any solution of the equation can be expressed as a linear combination of these two solutions,fl, f2. To obtain the odd and even solutions of ? 7 we must satisfy the following relations,

f(I) =f(r - E) for the even functions, and

f(E) -f( -(-) for the odd functions.

Our approximations do not distinguish between the values for ji corresponding to the even and odd functions, which are

f1i() -t f2(S) an even function, and

fl(.) -f2()) an odd function.

The Ionised Hydrogen Molecule.

By A. H. WILSON, B.A., Emmanuel College, Cambridge.

Communicated by R. H. Fowler, F.R.S.-Received December 19, 1927.)

1. Introduction.

The model which has been proposed for the ion of the hydrogen molecule H2+, consists of one electron and two protons. Since the mass of the electron is negligible compared with that of the protons, we may, to a first approximation, consider the protons as at rest. The system is then a particular case of the problem of three bodies, and can be solved completely classically. This has been done by Pauli,* and more recently by Niessen.t The value obtained by Pauli for the energy of the normal state is not in agreement with the experimental result inferred from the ionisation potential and heat of dissociation of the molecule. Niessen obtains the experimental result by the introduction of half integer quantum numbers.

* 'Ann. d. Physik,' vol. 68, p. 177 (1922), t 'Z. f. Physik,' vol. 43, p. 694 (1927).

2T 2

635 635



A. H. Wilson.

The classical problem is separable in elliptic co-ordinates, and so if we apply

Schrodinger's method to the system we shall obtain a wave equation which is

separable in the same co-ordinates. The resulting differential equations can be

solved exactly. This is the only three-body problem which admits of an exact

solution, and it is of interest to obtain an analytical result, and not merely one

obtained by a perturbation method, which it may be difficult to justify. In the present paper the differential equations defining the system are obtained

in ? 2, and their relevant properties stated in ? 3. In ? 4 we discuss the relation

between the equations, and obtain values for the energies of the various states.

Several surprising results appear. In the first place no solution is in general

possible. Solutions will only occur for certain distances apart of the nuclei.

In the second place it seems probable that these states are illusory, and that

there are no positive distances of the nuclei which give states. The difficulties

raised are examined in ?? 5 and 6.

2.1. The Differential Equations of the Problem of Two Fixed Centres.-The

ionised hydrogen molecule possesses nine degrees of freedom, and so nine co-

ordinates are necessary to specify its configuration. These may be taken as

the co-ordinates x, y, z of the electron mass m, and 1, vl, r1, 2, 2 22 ?of the two

nuclei both of mass M. If rl, r2 are the distances of the electron from the

nuclei and '12 the distance apart of the nuclei, then the wave equation for the

system is - 'J, 1 87t2/,8 12 e2 e2\

'8 E(y V%'i1 \2?23 + o ,T~ e

V.1 ,++ V,, +N . ..++ (2 +r r 2 r12'

where E is the energy. It is not possible to solve this wave equation completely, but it can be split

up into successive approximations* since m/M is very small. The zero approxi-

mation will be given by

1 V2r . 8~: 2 e2 M h 1 2 2 r(12/

To this approximation i only contains Ei, il, 1, i2, v2, '2 as parameters, and

for the purpose of solving the equation we can treat these quantities as fixed.

When the equation has been solved we obtain E as a function of the above six

parameters, and to obtain a stable state of the molecule we must choose them so

as to make E a minimum. This is equivalent to quantising the nuclei to a zero

approximation, and determines the possible distances apart of the nuclei.

Since rl2 is supposed fixed for the purposes of the calculation of ~ we need not

* M. Born u. J. Oppenheimer,' Ann. d. Physik,' vol. 84, p. 457 (1927).

636



Ionised Hydrogen Molecule.

mention explicitly the term e2/rlz. We can take it into account afterwards or consider it absorbed into the term E. In future we shall adopt that convention which seems most suitable for the purpose in hand. The three-body problem has now been reduced to a one-body problem, and we proceed to solve this restricted wave equation.

We consider a slightly more general problem than the ion H2+, and deal with one electron under the action of two nuclei with atomic numbers Z1, Z2, which are at a distance 2c apart. The electron is treated as a point charge.

We use elliptic co-ordinates

S (l + -- 2) /2C, = (ri - r2)/2c,

where r1, r2 are the distances of the electron from the nuclei. The third co-ordinate is the azimuth b round the line of centres. The ranges of the co-ordinates are

-1<n<00

0< <_ < 27.

The potential energy of the electron is

V =-e2 (Zl/rl + Z2/r2)

?2

C(~2 - ( 2) [(Zl + Z2) - (Z, - Z)],

V24 can easily be transformed to these co-ordinates by means of Gauss' theorem; and Schr6dinger's equation

V28 += (E - V) ~ - 0,

where E is the energy, becomes

8 {( 2 a a ?

@ 1) $- c 1

+ 8m2rGiE (02 _ ,2) + - {(Z1 + Z2) - (Z- Z2)} = 0.

4 must be finite, continuous, and single valued in the three-dimensional space i, 7, qb. The equation is separable and so we put

D= _ (~)X(()Y(7), therefore

( osin sin) 3

637



A. H. Wilson.

where n3 is a positive integer,

{(I - 2) + {X22 K'- +} Y , (2)

where X2 -= 8it22mcE/h2 (X > 0)

K := 7S2mce2 (Z1 + Z2)/h2 ., (3)

K = 8722e 2 (Z1- Z) /h2

J

, and X must be determined so that the equations (1) and (2) have finite and

continuous solutions in the ranges 1 X <oo, - 1 < < 1. 2.2. Before proceeding to the solutions of the equations (1) and (2) we can

obtain an insight into their nature by considering some special cases.

(1) If Z2 = 0 then K - K and equations (1) and (2) are identical. The

characteristics, or energies, are those of an atom with nuclear charge Zle and one electron - e. The solution X (i) is bounded for 1 < < oo . If it is also bounded in - 1 6 < < 1, then the same function of 73 is a solution of equation (2). This is actually the case, and so both at and X must be separately deter- minable from a single differential equation.

(2) If Z1i - Z2 1 we have K' - 0. This corresponds to the molecule H2+.

Equation (1) is the same as for the helium ion He+. The energy E may therefore take values which are included in those of He+. For such energies ,i and X will be determined by this equation alone. Equation (2) will give a further condition for ip in terms of X, and this condition, together with the two derived from (1), should determine both E and c.

The method we adopt is to determine pi so that equation (2) should have a finite solution, and also so that (1) has a finite solution. This will give two dis- tinct expressions for I, and the common roots ought to determine the possible energies.

In ?? 3 and 4 we apply the ordinary Schr6dinger theory, but do not obtain the experimental result. In ? 6 we lighten the restrictions on /, and then

obtain something near the experimental values,

3. Properties of the Functions X (X) and Y (n).

The equations take essentially different forms according as K (or K') is or is not

zero. If K is zero then the differential equation is that of the spheroidal wave

function, and offers no analytical difficulties. If K is not zero, solutions of an

638



lonised Hydrogen Molecule. 639

entirely different type are possible which are finite in the range 1 < < o .

These functions have been studied in detail by the author* with a view to the

special applications here. The general properties will be stated here, while

particular results will be quoted when they become necessary.

3.1. K 0 0, K $ 0.

We put X (t) = (2 _ 1)n3/2 e-^ y ().

The solutions which are finite in 1 < oo are of two types, and in both

y (i) consists of a terminating series. Those of the first type are also finite in - 1 < < 1, and the corresponding

solutions are suitable solutions for Y (t). n-1

y(~)= Z a~~. O 0

The conditions to be satisfied by X and ,u are that

K 2- = n + n3 (n >l)

where n is an integer, and t is given by a complicated function of X that can

be found for each value of n and which is of the nth degree in 1[. The second type of solution only exists when n3 : 0; and y (i) has a pole of

order n3 at i - 1. p-i

y(i) = (1+ S)- S E b "'. 0

In this case

=-P (p>l) 2X

where p is an integer and [, is a function of X of the pth degree in tp. The first type of solution is a possible one for Y (vi), but there are other types

possible. Y (7) may be an integral function of r, in which case there is a relation

between ,u and X expressed by means of an infinite continued fraction.

3.2. K = 0, K = 0.

If K = 0 there are no functions X ([) finite in 1 < < o .

If K' = 0, Y (T) must be an integral function, and p is given in terms of X by an infinite continued fraction. When X2 is small we can approximate to the roots of the fraction by series in powers of X2.

These results will be found set out in detail in the paper referred to. The

explicit forms of the solutions will be given when it is necessary to use them.

* A. H. Wilson, supra, p. 617.



A. H. Wilson.

4. Special Cases of the Two-Centre Problem.

We discuss three special cases: Z1 = 1, Z2 = 0; Z1 = 1, Z --- 1; Z -= 2,

Z2 = 1. The first is simply the hydrogen atom, while the second is the ionised molecule. The treatment of the atom is given to show the connection between the ion H2 and the ion He*.

4.1t. K - K'.

The equations for X (i) and Y (4) are identical in form. Solutions are given by

X () = (2 - Jl)/2 e-^ y (i), Y () - (1 - 2)j2 e-A y() where

n-1

Y ()= z ami . 0

The energy is determined by

t + 3 (n>1 ). (1)

If Z, = 1, Z2 = 0 this gives - E

(n + nf3)

where n is an integer > 1, R is the Rydberg constant. Ft is the same function of X for both equations, and a knowledge of its exact form is unnecessary. This is

just the usual result for the energy. The other types of solution do not give simultaneously solutions of both

equations, and must be disregarded.

4.2. K :f 0, K' = 0.

When Zl = Z2 this gives the most interesting case, the ion H2+. The equation for X (i) will be identical with that corresponding to the helium ion He+. Solutions will be given, as in ? 4.1 by K/2X = n + n3.

The function Y (T) will now take a different form. It may be either an even or an odd function. In both cases it must be an integral function. This will

give a relation between aL and X expressed by a continued fraction. We wish to determine the state with lowest energy, and so we take n3 = 0,.

and consider the even series for Y (q). The relation between a and X then is

1.22X2 3.4X2 5.6X2 - _ 2.3 2.3.4.5 4.5.6.7 .

~~~~~~~~~- ut~~~~~~ -- - - ------ ..(2t

2.3 4.5 6.7

4.21. The state with the greatest negative energy is given by n = 1.

640




The conditions arising out of the i equation are

c/2 == 1 and L.---= X2.

If we substitute = - X2 in (2) it seems probable that there is no real

solution except X = 0. The same is true if we use the odd series for Y (n). We

can prove this directly from the equation as follows. Consider the equation

d {(1 ^) _ 2(1 -2) y =0

the boundary condition being that y is finite at - =i 1. Then

dL )r dY '1 2) [dY ]

=Ldy- , _ - 1 2 + ?X2y2] d

and this cannot be true with X2 > 0. Therefore a necessary condition for Y (q) to be finite in - 1 < < 1 is that

( > - X2 when n3 = 0.

This solution which ought to lead to the lowest stationary state is therefore ruled out except when X 0=. The energy is then - E = 4Rh, the lowest

energy of the helium ion.

4.22. We now consider the state given by n = 2. The conditions arising from the i equation are

K

- =2and , 1- X2 l 1+ 4X2.

Since , > - ;2 is a necessary condition for the existence of Y (7) we must take the + sign in ,J.

If we attempt to find X from this condition and from (2), we find that there are no roots when X is small, but that the two expressions for J tend to equality for rather large values of X. It does not seem possible to determine for what values of X, if any, the two expressions for (J become equal. To do this we should have to tabulate , for large values of X, and the work involved in using (2) to a large number of terms would be prohibitive. An alternative method is to use an asymptotic expression for (. Such an expression can be obtained and is

JL -- X2 + 2X(r + n3 + 1) + O (1/X) (r an integer).

641



A. H. Wilson.

The term independent of X in this expansion is doubtful, and so no exact results can be inferred. All that can be said is that the two expressions for ,u tend to

-equality (with r = 0) as X becomes large. The corresponding solution may belong either to the molecular ion or to the hydrogen atom.

Two asymptotic solutions for Y (]) are given by

Y, ( e) ~A eos {(sin -lx)/(cos _x)r+'" O (1/X)} :and

Y, (7) ~ e-A Cos $ {(cos --x)T/(sin x)r2+0 (1/X)},

where 7 = cos x and we are supposing n3 = 0.

AlsoY1(]N)+Y2(7) isanevenfunction of q, and Y1T ()-Y2 () isan odd function. If we use either of the last two functions we shall obtain a state of the molecular

ion, as Y (7) is then large near v = - 1 and small elsewhere. If we use Y1 (r) we then have a wave function which is only apparent near =- 1; and Y2 () gives a wave function apparent near - = 1. These last two solutions

;correspond to a hydrogen atom perturbed by a very distant nucleus. In

these types of stationary states the electron is in an orbit near one nucleus for a long time and then passes over to the other nucleus for the same period. Such ,orbits are of no interest for the ionised molecule.

We cannot actually decide whether there is a state of the ion corresponding to this solution, but even if there is, it cannot give the experimental result for the lowest energy.

Let us denote the energy of the electron by Eel. Then there is also the

.energy due to the repulsion of the nuclei. This is e2/2c. The total energy E = Eel + e2/2c. If I is the ionisation potential of the molecule H2, then - E + I is the total work to be done to remove all the components to infinity. This work is also D + 2Rh, where D is the energy of dissociation of the molecule into atoms. Therefore

I- Eel- e2/2c = D + 2Rh.

In the particular case we are considering - Ee = Rh. Therefore I = Rh D e2/2c.

D is known to be about 4 * 4 volts. This gives I a lower limit of 18 volts however

large c may be. There seems to be general agreement that I is just less than 16 volts, and so even if the state exists, it cannot give the experimental result. Also any value of K/2X > 2 will give a still higher value of I and deviate still further from the required value.

4.23. The investigation for higher values of n is difficult on account of the

complicated relations for i. It is certain, however, that no solutions exist for small values of X.

I642




The other types of solution for X (~) furnish no values of p common to the two

,equations. It therefore seems probable that the system admits of no stationary states.

4.3. K - K $ 0.

We consider the two-centre problem given by Z1 = 2, Z2 = 1. Since the

equations connecting ,t and X will be algebraic equations, we ought to be able to decide definitely the existence or non-existence of stationary states.

X (i) and Y (q) will consist of the product of e-h^ or e-^" and a terminating series. To find the lowest state we take n3 = 0 and make - E a maximum.

The largest value of - E for which the equation for Y (]) admits of a solution is given by K'/2X 1. This necessitates K/2X = 3. We then have in addition two relations between X and p.

From the equation for X (i) we have

32X2(t1 +- 2) - 4X(3Xt12 + 16t1 +- 12) +- , + 8122 + 12p1 = 0,

where ~- = - - X2 + 4?X.

From the q equation we have - X2 or l, = 4X.

Eliminating i1 from these equations we have

X -= 0.

The common roots are therefore X = 0 and X = oo. Neither of these values

gives a solution of the two-centre problem. It is not possible to treat the

general case in this manner, but it seems probable that no solutions exist. 5. From the results of the previous sections it is only possible to draw the

conclusion that it is not correct to treat the problem of two centres in this manner. The question then arises what attitude we are to take towards the

Xdifficulty. Several attempts have been made to determine the states, and we shall examine these.

Burrau* used the equations of ? 2.1 and obtained their characteristic numbers

by numerical integration. Equation (2) presents no difficulties either in the numerical method or in the exact analytical investigation. Equation (1) presents many difficulties from the analytical side, and is not satisfactorily solved in Burrau's paper. His method is to transform the equation into a Riccatti equation and then to solve this numerically. Inhistreatmentof equation <1) he expands the Riccatti equation round the irregular singularity at infinity. This expansion will in general be divergent, but nevertheless asymptotic. To make it convergent will require an extra condition to be imposed on (t. Burrau

*' K. Danske Vid. Selskab.,' vol. 7, No. 14 (1927).

643



A. H. Wilson.

does not justify the use of the asymptotic series, and it is probable that this invalidates his solution. He obtains the experimental result for the lowest

energy; but his equations are equivalent to those used here, and the analytical solutions show no results corresponding to his.

Hund* attempted to obtain qualitative information about the term spectrum of the two centre problem by considering the transition from infinitely distant nuclei to coincident nuclei. To do this he considers a one-dimensional problem, and then generalises it to separable systems. The equations of the two-centre

problem are not of the type he considers. However, he only uses these results to prove that, during the passage from infinitely distant nuclei to proximate nuclei, no terms are lost. We can prove this for each of the equations (1) and

(2) by means of their asymptotic expansions for large X. The form of the solution of (1) is independent of the value of X, and cannot be affected by the transition. The corresponding result for equation (2) has already been proved in ? 4.22. When X is large the electron may be in the neighbourhood of either nucleus. The argument in ? 4.22 shows that when X is small we shall have two characteristic functions, one even and one odd, corresponding to each state when X is large. The states which exist for large X must be given weight 2, as the electron has two possible orbits of equal energy round the two nuclei. The number of states, therefore, remains the same.

Uns6ldt has obtained the energy of the normal state by a perturbation method. He considers a hydrogen atom perturbed by a proton and calculates the additional energy by successive approximations. The difficulty about this method lies in the convergence. Unsold finds that the second approximation is nearly twice as large as the first, and has the opposite sign. Neverthe- less he obtains the experimental result, but does not discuss the third

approximation. In the previous sections we have seen that the solutions X (i) which are finite

in 1 < i < oo do not give any states of the system. This is due to the fact that two conditions are necessary to make X (i) finite, whereas we should expect that

only one condition is necessary. We should expect the electronic motion to contribute two restrictions, one from each of the equations X (i) and Y (n).. We would then be able to quantise the nuclei by making the total energy a minimum. But since the electronic motion furnishes three restrictions, there are no parameters available for quantising the nuclei. It is also suggestive that Burrau has obtained good agreement with experiment by imposing only two conditions.

* ' Z. f. Physik,' vol. 40, p. 742 (1927). t 'Z. f. Physik,' vol. 43, p. 563 (1927).

644



lonised Hydrogen Molecule. 645

If we allow such a procedure we must abandon our restriction that 4 must be finite throughout the whole space.

In the next section we deal withwavefunctions which are not finite everywhere.

6. The Energy of the Normal State.

If we allow ~ to become infinite, we have a much larger class of functions to deal with. If 4 is to have any physical significance it seems necessary that it should vanish at infinity. We do not, however, demand that $ should be finite at = 1.

Such solutions are best obtained in descending powers of the variable. If we put

X^ (i) =^ - )"3l e-^" y() and

t= (1 + r)- then

y (t) = t"13+1-/2 E a tm, o

where

{( 3 irts-nj m.2X)A 2 1)

+ (n3 + 2X) (n + 1) -- _ - ~ 2 ai1

-2a (m2m+n3-1 )(m-l-_)=o. (1)

We can write this in the form

Nm-l + ur-1-- v,,2 Nm =2 = ?02 where

Nn = qa,1

If the series is to have a non-zero radius of convergence we must choose A so as to make L N, = 2. This condition can be expressed in the form of a con-

-

tinued fraction

N1a = a l/ao= - -= vo (U1 + N,) v .... (2) u1 + U2 + u3 +

This infinite continued fraction is convergent, and so we can use it to determine the values of p. which give a solution of the required type. Although this solution is finite as [ -c oo along the real axis, it is not finite at [ - 1. The second approximation to Nm is

Nm2 {1-(1 - s)/Im}.



646 lonised Hydrogen Molecule.

Near i = 1, X (i) behaves like ( - l)-n3/2 or like log (- - 1) and so does not- fulfil the condition of being bounded. In spite of this lack of boundedness we shall use this type of solution, and calculate the energy it gives for the normal: state.

The continued fraction (2) is in the form adapted for the calculation of the smallest characteristic number, and on putting n3 0 it becomes

tJ+2pX+ X2-2 x+(1-p) (4X+p) 2x

2 (l-p)2 8 (2- p)2 (3

(2-p) (1-i-4-p) +2X-t--2p.--X2+ (3-p) (2-4X--) +2X--2pX-X+2- . where p == K/2X.

We can obtain another relation for V from ? 4.2 equation (2), arising out of the equation for Y(^,). This second relation is also in the form of an infinite, continued fraction. In this case, however, we can easily approximate to the' roots by expressing Vp as a series in powers of X2. The lowest root is given by

1 2_ 2 X4 4 16 XS37.5 (4)7 =

3 135 35.7 37 .53 72

These two expressions (3) and (4) determine p as a function of ;. It is possible; to approximate to (3) by a series; but the expansion is of little value, as the~

convergence is slow owing to the presence of the terms involving p. At this

stage it is best to adopt a numerical method. If X is not too large we can obtain an accurate value of pz from (4) and substitute this in (3). For any given value of X an approximate value of p can be obtained by inspection, and then the successive convergents of the fraction can be calculated, and the error can be

roughly determined. By several trials p can be calculated accurately. Using this method a few values of p have been rather roughly computed, and the' values obtained lie quite close to those obtained by Burrau. Although these, results do not agree entirely with Burrau's, the agreement is sufficient to make it fairly certain that it is the present type of function that Burrau has used. Burrau obtains values of Eei for various values of c. He then adds the nuclear- energy e2/2c and then determines c by making this total energy a minimum.. Burrau's method gives a value for the energy which, considering the complicated' motion of the nuclei, is in good agreement with the experimental result. We'

may therefore safely assume that the present method leads to the correct result. The actual calculation has not been carried out, as it would not lead to a result

materially different from Burrau's, and also because 4 does not satisfy the- condition of being bounded.



Band Spectrum of Water Vapour. Band Spectrum of Water Vapour.

It is difficult to see what the physical significance can be of a wave functiona which becomes logarithmically infinite. The infinity is certainly not a very large one, but for states in which the rotational quantum number is not zero the wave function has a pole of order n3/2. A similar difficulty arises in the-

theory of the relativistic hydrogen atom. In this problem inmvolves a factor r-A where X is a small positive constant, and d is infinite at the nucleus. In neither problem is there any obvious interpretation of the infinity.

In conclusion I should like to thank Mr. R . H. Fowler very much for many- helpful discussions and criticisms.

The Band Spectrum of Water Vapour.-II.

By DAVID JACK, M.A., B.Sc., Assistant and Carnegie Teaching: Fellow, The

University, St. Andrews.

(Communicated by 0. W. Richardson, F.R.S.-Received February 17, 1928.)

1. New Measurements of the Band X 3428.

In a previous communication* on the band spectrum of water vapour refer-

ence was made to some measurements by Tanakat on the band 3428. The author

has since measured and analysed this band. The experimental arrangements: were the same as for the work on the band 2608. Two plates exposed for

about one hour were measured, and the results showed close agreement. The

dispersion obtained, 8.0 A. per millimetre, was inferior to that in the band

2608, but the consistency of the measurements indicates that errors are unlikely to exceed 0 05 A., while the probable error is considerably less. Overlapping: of lines is somewhat more troublesome in the band 3428, and consequently there^ is a slightly increased tendency to irregularities in the results.

Details of the band are given in Table I, arranged as for the band 2608..

Some fairly strong lines present on the plates are not accounted for in the table..

Comparison of the term differences in this band with those in the bands

previously analysed shows that the initial terms are the same for the two bands, 3428 and 3064, and the final terms are the same for the three bands 3428, 3122 and 2875. This is in agreement with the scheme already given.:: The following;

* Roy. Soc. Proc.,' A, vol. 115, p. 373 (1927). t ' Roy. Soc. Proc.,' A, vol. 108,. p. 594 (925), L Loc. cit.

It is difficult to see what the physical significance can be of a wave functiona which becomes logarithmically infinite. The infinity is certainly not a very large one, but for states in which the rotational quantum number is not zero the wave function has a pole of order n3/2. A similar difficulty arises in the-

theory of the relativistic hydrogen atom. In this problem inmvolves a factor r-A where X is a small positive constant, and d is infinite at the nucleus. In neither problem is there any obvious interpretation of the infinity.

In conclusion I should like to thank Mr. R . H. Fowler very much for many- helpful discussions and criticisms.

The Band Spectrum of Water Vapour.-II.

By DAVID JACK, M.A., B.Sc., Assistant and Carnegie Teaching: Fellow, The

University, St. Andrews.

(Communicated by 0. W. Richardson, F.R.S.-Received February 17, 1928.)

1. New Measurements of the Band X 3428.

In a previous communication* on the band spectrum of water vapour refer-

ence was made to some measurements by Tanakat on the band 3428. The author

has since measured and analysed this band. The experimental arrangements: were the same as for the work on the band 2608. Two plates exposed for

about one hour were measured, and the results showed close agreement. The

dispersion obtained, 8.0 A. per millimetre, was inferior to that in the band

2608, but the consistency of the measurements indicates that errors are unlikely to exceed 0 05 A., while the probable error is considerably less. Overlapping: of lines is somewhat more troublesome in the band 3428, and consequently there^ is a slightly increased tendency to irregularities in the results.

Details of the band are given in Table I, arranged as for the band 2608..

Some fairly strong lines present on the plates are not accounted for in the table..

Comparison of the term differences in this band with those in the bands

previously analysed shows that the initial terms are the same for the two bands, 3428 and 3064, and the final terms are the same for the three bands 3428, 3122 and 2875. This is in agreement with the scheme already given.:: The following;

* Roy. Soc. Proc.,' A, vol. 115, p. 373 (1927). t ' Roy. Soc. Proc.,' A, vol. 108,. p. 594 (925), L Loc. cit.

647' 647'



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The Ionised Hydrogen Molecule