1552
R. E. GLICK
Vol. 65
O X THE DIAR.IAGKET‘IC SUSCEPTIBILITY OF GASES’ BY R. E. GLICK Department of Chemistry, The Florida State University, Tallahassee, Florida Received March 16, 1961
The diamagnetic Susceptibility of He (xae) has been calculated using 14, 18 and 20 term Kylleraas wave functions. Employing certain assumptions, these values are plotted against the corresponding energies. x H e , corresponding t o that which would be obtained using a wave function with an infinite number of terms, is ( - 1.8914 =k 0.0002) X This value is recommended as a standard in the experimental determination of the diamagnetic susceptibility of gases. An analysis of experimental data is performed, most probable values of diamagnetic susceptibilities of gaseous elements are obtained, and this analysis is used to obtain most probable susceptibilities for gaseous hydrocarbons.
Interest in theoretical and experimental aspects of molecular magnetism has reappeared, based seemingly on details connecting molecular magnetism and nuclear magnetic resonance spectroscopy, and catalyzed by the rich variety of applications to molecular theory and analysis issuing from nuclear resonance studies.2 The basis for this relationship centers on electronic effects observed when 5~ molecular system is perturbed by a magnetic field. Molecula,r diamagnetism is related to magneto-electronic effects as seen by an observer external to the molecular system, while nuclear resonance spectral shifts arise from magneto-electronic effects as seen by an observer residing at a particular site within the molecule. I n discussing elements relating these two experiments, some difficulties, characterized by lack of precise experimental data, exist. As for example, theoretical calculations of the diamagnetic susceptibility for the lower hydrocarbon gases such as methane, ethvlene and acetylene have recently a ~ p e a r e dand ~ , ~ seem to be capable of high precision; the need for reliable experimental values is obvious. Not until very recently, however, have experimental values for these quantities6 been obtained, and, to a certain extent, the values so determined need more adequate referencing. I n this connection, there is a need for a suitable standard in experiments designed to investigat,e the diamagnetic susceptibility of gases. This difficult,y appears t o be related to the nature of present experimental met hods. I n the several experimental methods6-10 that have been employed for determining the diamagnetic susceptibility of gases, the st,andard reference value has been the paramagnetic suscept,ibility of oxygen,.~ 0 % In . general such methods are ratio methods with responses that are directly proportional to the values measured. Since the mag-
nitude of the susceptibility of a diamagnetic gas will be O.O!th or less than oxygen’s,sensitivity is severely taxed. Although experimental determinations of xor do not agree,l‘ as will be discussed later in this paper, no serious difficulty arises since a reasonable choice, based upon theory and experiment, can be made. It is the purpose of this paper to introduce the value for the diamagnetic susceptibility of helium, XH?, determined by calculation rather than experiment, as the primary reference. Presently available quantuni solutions are such that only for helium can high precisioii (one part in a thousand or less) diamagnetic susceptibility calculations be made. Exact ground state wave functions are available for helium12and such wave functions axiomatically reflect those properties of the diiute gas that can be obtained from them. That only ground state wave functions are required in this calculation can be seen from an examination of the theoretical expression for magnetic susceptibility. The value for the magnetic susceptibility of any chemical species (xm,molar) can be obtained from a three term expre~sion’~;a temperature dependent term and two temperature independent terms
(1) Support by a grant from the Petroleum Research Fund, American Chemical Society, is gratefully acknowledged. ( 2 ) See f o r example, Chapt,ers 2, 3, 5 and 7 in J. A . Pople, W. G. Schneidor and 13. J. Bernstein, “High Resolution Nuclear Magnetic 1959. Resonance,” McGraw-Hill Book Co., Inc., New York, N. Y.. (3) I n particular see J . Guy and J. Tillieu, Compt. rend., 942, 1279 (1936); J . Clwm. P h y a . , 24, 1117 (1956); R. E. Glick and D. F. ICates, ibid., 33, 308 (1960). (4) W. Wenter, ibid., 28, 477 (1958). ( 5 ) C. Barter, R. G. Meisenheimer a n d D. P. Stevenson, J . I ’ h ~ s . Chem., 64, 1312 (1960). ( 0 ) For general reference t o experimental methods see: P. W. Solwood. “Magnetochemietry,” Interscience Publishers. Inc., New York, N. Y . , 1956, 2nd Ed., Chapters 1-4. (7) G . G. Ilavens, Phys. Rea., 43, 992 (1933). iri) IC. E. Mann, Z. Phguik, 98, 548 (1926). !Sj A. P. Wills and I. G. Hector, Phys. Reu., 23, 209 (1924). (10) T. S o d , Phil. Mag., 29, 305 (1920).
in order to determine the diamagnetic susceptibility of a substance such as a noble gas. The co~istants’~ for the third term in (1) are; Avogadro’s number ( N ) , the electric charge ( e ) , electron mass (m), and velocity of light (e). Also,
The first term, the leading term for paramagnetic substances, depends strongly on the number of unpaired electruns in the atom or molecule. The last) two terms, important for substances with paired electrons, are the so-called paramagnetic or high frequency contribution to molecular diamagnetism and the classical diamagnetic or Larmor term, respectively. For substances with paired electrons the first term is zer0.1~ For atoms with paired electrons the second term is also ze1’0.l~ It is necessary, therefore, to evaluate ($ Ti2$) i
(11) Ref. 6, p. 245 f. (12) C. 5. Pelreris, Phys. Rev., 112, 1649 (19581, describes a function determined by t h e variation method which yields a non-relativistic
Sea also J. C. Slater, “Quantum energy acaurate EO 0.01 em.-’. Theory of Atomic Structure,” McGraw-Hill Book Co., Inc., Xew York, N. Y., 1980, Vol. 11, p. 36 f. (13) J. H. Van Vleck, ’ T h e Theory of Electric and Magnetic Susceptibilities,” Oxford University Press, London, 1932, Chapter 10. (14) Values are thosa due t o Bearden and Thomson taken from C. D. EIodgman, Editor-in-Chief, “Handbook of Chemistry and Physics,” Chemical Rubber Publishing Company, 40th Ed., p. 3332.
1553
DIAMAGNETIC SUSCEPTIBILITY OF GASES
Sept., 1961
TABLE I WAVE F I J S C T I O N COEFFICILNTS
14
14
Terms Term 1
$1 $1.03474 x + I ,19705 X 1-3 08547 X + I 37733 x -8.09208 X t5.21399 X -1.48873 X +3,94007 X $6 61307 X +7 51844 X $2 14655 X -3 08377 x --9 38369 X
U
t2 S S2 U2
SU t2U
Ua
t 2 u2
st= Sa 1 2 ~ 4
u4
3-1
3-1 +1,05602 X + I 24326 X -1 09544 x + I 77168 X -8 04649 x $ 3 84558 X -1.41782 X +1.16680 X +6.54696 X $ 5 71026 X - 2 38856 X -3 69166 X -1.02680 X
10-1
loW3 10-3 IO-' lW3
13-4 10-6 10-
IO4
20
18
US t2U3
6*t2
s4
10-I
IO-* 10-3
loe4 10-j 10-5 lo-*
+ I 07607 X +1 43009 X -+7 53535 x 1.3 40002 X -I 00588 X $5 33987 X -2 20582 X $7 96309 X + I 98073 X 1-4 94663 X + i 2509:3 x $1 54382 X -4 52176 X $1 10026 X -8 897YQ X 1 - 1 39961 X -3 66743 X
$1
+1,07770 X +1.39125 X - 1,35267 X +3.27772 X - 1.04978 X $5.49697 x -2.17180 x +8.22586 X +1.81478 X f5.50529 X -1.63861 X fl.32497 X -4,11474 X $8.13960 X -8.35317 x $-1.026Y1 X +6.09189 X
10-l 10-3 10
lou4 10-4 lou7 lo+' lo6 10 lo-' lo-'
'
lo-'
10-3 lo-' 10-3 10-3
lo-' loe4 lo-' 10 -7 lo-'
10 - 6 lo-'
+7.157L5 X 10."
SPU
$3.38258 1.36686946 x lWfi
14
1.3617172
7
1.3633714
is the radius of the ith electron, and the sum is carried over for all electrons. The wave functions employed for the purpose of determining E62 are of the Hylleraas15type as ri
i
given by
,k
=
,,e-ks/Z
zClmm k(1fmi.n)
In the calculation of
(~rz
r,2,$)
all of the in-
i
81 t m
un
(2)
tegrals are easily evaluated and are of the form
l'or equation 2, s, t and u are functions of rl, r2 and r12 the coordinates of the two electrons and their distances from one another and are given by The values for the diamagnetic suscept.ibilit'y of helium ( x H ~ ) obtained by employing the 14, 18 s = rt rz, t = r2 - rl, u = r12 and 20 term wave functions are given in the second Hylleraas wave functions as determined by column of Table 11. Chandrasekhar and Heraberg and co-workers16-18 TABLE I1 have been employed in this study. Those used T H E D I A M A G N E T I C SUSCEPTIBILITY AND EXERGY O F €It? AS contain 14, 18 and 20 terms. The combinations OBTAIXEDFROM VARIOUSHYLLERAAS WAVEFUNCTIONS of 8 , t and u for these wave functions are given in XmO E ( X 108) (a. u.) kin kout the first column of Table I. Rather than determin- Terms ( x x ' i O 3 ) 14 -1.8888 -2.9037006' 3 , 8 5 = 3.849930IG ing Clmn, these workers tabulated ~ l , , . k ( ~ f ~ + n ) . 14 -1.8895 -1.889 -2.9037009R 3.75" 3.7500555' For our purposes, clmnis more useful. These are 18 -1.8896 -1.890 -2.90370634 3.85" 3.8499613" also shown in Table I. The elect.ron radii (atomic 20 -1.8912 -1.8912 -2.9037179b 3 . 8 7 * 3.8699977b units) in terms of the Hylleraas variables are given m -1.8914 -2.9037237'
+
(1.5) .J. H. Rartlett, J. J . Gibbons and C . G. D u n n , P h y s . Reu., 47, 679 (1935), have shown t h a t Ilylleraas wave functions cannot be "formal" solritions t o t h e Schrodinger equat,ion since recurrence relations for t h e SchrBdinger equation in this caae require t h a t clmn 3 0 for all I , m, n. T. Kinoshita, Phys. Ren., 106, 1490 (1957), in examining this question, derived a set of functions, a = 8, p = u / a , and q = t / a and constructed a n e w wave function
$, =
?e-8/2
clmn 81 prrz q~ =
.,,e-*iZ
p l m n - l sl-m
tLm-n
tn
which is a formal solution t o the SchrAdinger cqiiation. This set, containing t h e Rylleraas function a s a subset and, in addition, negative powers of the variables, satisfies t,he formal solution requirements for t h e Schrodinger equation. These requirements have been given by T. Kato. Trans. Am. Math. SOC..7 0 , 195, 212 (1951). Such requirements do not preclude t h e use of Hylleraas functions as sufficient solutions for our purposes, as these functions appear t o converge, if energy is to be t h e criterion, with sufficient rapidity. This conclusion could not be dismissed by Kinoshita. (16) S. Chandrasekhar, D. Elbert and C . IIeraberg, Phys. Reo., 91, 1172 (1953). !17) S. Chandraselchar and G. Hereberg, ibtd., 98, 1050 (1955). (18) J. F. Hart a n d G. Iierzberg, ibzd., 106, 73 (1957).
aRef. 17. b Ref. 105,1490 (1957).
18.
T. Kinoshita, Phys. Reu.,
Before discussing these solutions, t'he quality of the wave function must be considered. A measure of quality would be as follows17: the coefficients for these wave functions are obtained by the variation method except for the exponential constant, IC. In principle, the best value for the energy that may be obtained for a given number and selection of t'erms will be found when the k used in the trial function equals that obtained by relations defining IC as a function of the coefi~ients.'~Thus k is not an independent variable, and the best solution must be determined by an iterative process. As this process required extensive computational effort, even with high-speed (19) Not only are k andE functions of t h e coefficients, b u t in the general solution k = 2 d - E . See ref. 12.
B.E. GLICK
1554
TaBLE
111
I)IAX.kGNETIC SUSCEPTIBILITIES O F SOME
He
Ar
Ne
Vol. 65
GASEOUS ELEMENTS ( x lo6) Kr
Xe
H2
ti2
Manna -6.70 -19.4 -27.8 -42.23 -4.11 -12.24 Havens" -1.95 -7.85 -19.7 -4.2id -12.8" Hector' -2.04" -7.21' -19.7k Itoth" -19.7 Abonnenc' -19.7 -29.9 -45.2 Soneg -4.10 BMSh -2.05 -7.08 -19.tjk -29.5 -45.9 -4.16 -12.5 -29.7" -45.5" Average' -7.21 -2.01 -19.6 -4.04 -12.4 Most probablem -29.6 -45.4 -1.89 -7.10 -19.5 a K. E. Mann, 2. Physik, 98, 548 (1936). * G. G. Havens, Phys. Eeu., 43, 992 (1933). c L. G. Hector, ibicl., 24, 418 (1924). A. P. Rills and L. G. Hector, ibid.? 23,209 (1924). e W.Gerlach and A. Roth, 2. Physilc, 85, 544 (1933). L. Abonnenc, Compt. rend., 208, 986 (1939). g T. Sone, Phil. Mag., 39, 305 (1920). h C. Barter, R. G. Meisenheimer and D. P.Stevenson, J . Phys. Chem., 64,1312 (1960). This is the standard value used in BMS and Hector studies and is the averagr of x.kr from other studies. Arithmetic mean. -4verage minus -0.12 x 10-6. n illann's values not included. f
A plot of the diamagnetic susceptibility versus the -1892
I
___---
-I 888
-2903705
-2903710
- 2 903715
-2903720
E ( A UJ, Fig. 1.-The molar magnetic susceptibility of helium using "best quality" 14, 18 and 20 term Hylleraas u a v r functions (left to right) versus the total electronic energy as obtained from these functions. The dotted extrapolation terminates at the non-relativistic limit expected from this type of wave function
computers, reasonable choices in k are usually made. The best quality wave function for a given set of Irariables, therefore, mill be obtained when k17 equals koUtas well as when the energy has been minimized. I n this connection, reference may be made to the appropriate columns in Table I1 for the 14 term wave function. For this case, the best wave function will be bracketed such that kLn will lie between 3.85 and 3.75. This is based upon the relationship between k,, and kOutsuch that for the higher choice IC,, is less than and the converse is true for the lower choice. The best quality wave function, however, mill yield an energy that is lower than that in either case. In treating the magnetic susceptibility data of Table 11, we make the following assumptions: ( I ) the best quality wave function will yield a magnetic susceptibility, Table 11, column 3, which varies continuously and monotonically with energy; ( 2 ) this value will yield upon extrapolation to infinity, where infinity refers to the number of terms in the expression, the best value for the X H ~ . For this extrapolation the energy value for an infinite series of terms is that of Kinoshita obtained by employing a similar extrapolation. These \-slues are given in Table 11, as m entry.
energy is presented in Fig. 1. To the extent that t8his extrapolation is valid, the limiting molar diamagnet'ic susceptibility for helium is - 1.8914 X lov6 f 0.0002 (chemical scale). There is no a priori reason to expect that a function derived from a wave function determined by the variat'ion method mill yield an observable varying in the same way as does the energy. Kevertheless, on examining Fig. 1, it appears, even with the limited data, that this relationship exists. Although many calculationsz0of X H ~have been performed, none have taken advantage of the complete wave functions that have been employed in this study. The value herein obtained should be accurate to the indicated error *0.017,, with the error most probably associated with the knowledge of the physical constants used in the evaluation of the last term in (1) ra,ther than uncertainties in the wave function. Although experimental values for X H ~might be interpreted as consistent with t'he calculated value, a closer examination reveals some decided inconsistencies. The values that have been obt'ained experimentally are: Havens,' - 1.91 X Hector,21- 1.88 x 1 0 ~ and ~ ; most recently, Barter, Jleisenheimer and Stevenson,5 (BMS), -2.02 X The precision in the latter study is +4%, and less than 1% for t'he other two. In order to make a direct comparison between these various values, however, it, is necessary to adjust them to a common basis. This is presented in Table 111. I n Table 111, all of the det,erminat,ionsof the diamagnetic susceptibilit'y for gaseous subst,ances have been collect,ed, and have been adjust,ed by t'hree self-consistent mays. (1) I n t'he atudies of Llann,8 Havens' and xo2 has been wed as a st'andard. The values entered in Table I11 h a w and experibeen normed to the best, t8heoret,ica122 mentalz3 value for the magiiet,ic susceptibilit,y of 3.42 X lov3 (20'). (2) Abonnenc used oxygen, the susceptibility of hydrogen as his standard, and his values are normed to XH) as found in Table
-
+
(20) For a compilation of t>hesesee ref. 6. p 7 2 , using reasonable wave functions. susoeptib e8 in the range (-1.6 t o -2.0) x are obtained. (21) L. G. Hector, P h y s . R e i . , 24, 418 (1924). (22) RZ. Tinkham and 11.n-. P. Strandberg, i b i d . , 97, 937 (1955). (23) R P f . 1 3 , 11. 266 f .
Sept., 1961
DISMAGNETIC SUSCEPTIBILITY OF GASES
111. (3) The values obtained by Barter, Aleisenheimer and St,evenson (BMS) and Hector are normed t o x.kr as adjusted above. The value of xAkr due to Gerlach and RothZ4 was obtained by an absolute method and has not been adjusted. The average values are simply the mean. As is seen from Table 111, the average experiThis is mental value for x H e is -2.01 X considerably different from the value herein calculated. As the calculated value is most probably correct, -within the indicated limits of error, the average experimental value must be incorrect. Probable values are listed in Table 111, as t'he final entry, and are obtained by applying a correction based on the assumption that the response, under the various experimental conditions employed, would be the poorest for the lighter and least diamagnetic element's. This is consistent with error, for example, in t'he study of Barter, Meisenheimer and Stevenson and as pointed out by Gerlach and. RothZ4for Havens' experiment. Both of these studies used essentially the same experimental method. This adjust'ment is made by assuming an absolute error of -0.12 X low6, and subtracting this quantity from all values. I n this may, X H ~is adjusted to the value calculated jn the study. x+ as corrected is found to agree with that obtained by a direct calculation using the 11 term James and Coolidge wave function for the radial ) ~ last term of part of the susceptibility, ( ~ 1 1 ~the equation 1; the value2j is -4.15 X To this is added the value for the second term26 of (l), 0.0846 x loF6as determined from molecular beam experiment^.^^ Thus, XH? theoretical, equals --4.07 X while the experimental value is --4.04 X lo+. I n addition, it is of importance to re-examine the values obtained by BMS for obher gaseous substances. The correct'ion is simple multiplication of the various x's by 19.6/19.3, the ratios of the scale values, xhr, used in this study as compared to that of BMS. These values are recorded in Table IV. In addition, ACH2, the increment in magnetic susceptibility for CH2's in homologous and compares series; is adjusted to -11.6 X favorably with t>hatof -11.68 X in an analysis of the susceptibility of liquids.
+
(24) T. Cerlacli and A. Roth, Z. P h y s i k , 85, 344 (1933). ( 2 5 ) E. E . Wittmer, Phus. Rev., 61, 383 (1937). (26) I. E w e , ibid., 103, 1254 (1956); see also N. F. Ramsey,
"Moleoolar Beams," Oxford University Press, New York, N . Y . , 19513. (27) For 8, dimussion of other calculations see W. Weltncr, 3. Chem. r h u s . , 28, 477 (1958).
1555
TABLEIV DIAMAGNETIC SUSCEPTIBILITIES OF SOMEGASEOUS MOLECULES
Hydrocarbon
Gasa
Liq.
Methane Ethane Propane %-Butane %-Pentane Isobutane Ieopentane Neopentane Cyclopropane Cyclobutane Cyclopentane Acetylene Ethylene Allene Propylene 1,2-Butadiene 1,3-Butadiene 1-Butene Isobutylene
-17.6 -27.1 -39.1 -51,o -50.0" - 6 2 . 4 -63 3" -63.05* -51.2 -51.7' -63.9 - 64.40' -63.9 -63.1' -39.7 -40.5 -57.0 59. 2b -21.0 -19.0 -25.6 -31.1 -31.5b -36.1 -32.5 -41.6 -41.3 -44.4b ACHz -11.6 b S . Broersma, J . Chem. Phys., 17, 873 (1'349). 5 Ref. 5. J. R. Lacher, J. W. Pollock, W. E. Johnson and J . D. Park, J.Am. Chem. Soc., 73,2838 (1951).
Furthermore, the values obt'ained for a given substance in the liquid phase as compared to the vapor phase agree more closely. This correspondence, that the diamagnetic susceptibility is independent of phase, seems to be general for nonpolar substances. Susceptibility measurements on benzene, for example, quite closely exhibit this phase independence. Benzene might be expect'ed to show ext'reme behavior, based on the decided magnet'ic anisot'ropy found in this system.29 In summary, t,his re-examination of the diamagnetic susceptibility of gases is based, provisioiially, on three theoretical x's: those of oxygen, hydrogen and helium. It is felt that if more sensitive equipment were to be employed in the determination of diamagnetic susceptibilit,ies, t,he absolute standard, X H ~ ,could be employed. This would require an a,pparatus approximately t,wo orders of magnitude more sensitive than existing equipment. Such equipment would not be suitable for measurement, of xoz but cert,ainly sufficient for all diamagnetic gases. (28)
W.R . kngu8, G. I. W. Llewelyn and G , Stott, Trnns. Piiroday
R . Angus, F. R . Hollows, G. Stott. 1). 11. Iihanolkar and G. I. W. Llewelyn, ibid., 66, 890 (1959). (29) J. Hoarau, N. Lumbroso and A. Pacault,, C o m p t . r e u d . , 242, 1702 (1956).
Soc., 55, 887 (1959); W.
