4537
FACTOR ANALYSISOF SOLVENT SHIFTSIK PROTON MAGNETIC RESONANCE ing a connection between the nature of the substituent and the symmetries of the affected orbitals. Experimental Section Anhydrides 1 and 2 were prepared from a-pyrone precursors by a method to be described in detail elsewhere. l7 Nmr spectra were recorded on a Varian Associates HA-60-IL spectrometer operating at 60 MHz in the frequency swept mode. Chemical shifts were measured directly from spectra traced on the 500-Hz scale at a sweep speed of 1.0 Hz/sec and are judged to be accurate to f 0.03 ppm. Coupling constants were read directly from spectra traced on the 50-Hz scale at sweep speeds ranging from 0.1 to 0.02 HZ/sec and are Precise to 0.03 Hz. Inaccuracies introduced by errors in chart calibration are certainly less than 2%. Decoupling experiments were done while operating in the frequency swept mode, using either a Hewlett-
*
Packard Model 200AB signal generator or a General Radio Model 1304B audiooscillator. Because of the small sizes of the long-range coupling constants, components of multiplets were not always well separated (cf. the resonance of H(c) in Figure IC),so that it was not always possible to irradiate only one part of the multiplet. In these cases, a series of irradiations was done, beginning at the low-field side of the pattern to be irradiated and moving upfield in ca. 1 Hz increments until the high-field side of the multiplet was reached. Although it was not possible, under these conditions, to effect total selective spin decoupling, the patterns of skewing of affected multiplets sufficed to define the relative signs. Acknowledgment. Partial support of tkis work: by a National Science Foundation research grant (GP-11107) is gratefully acknowledged. (17) G. Vogel and D. J. Sardella, to be published.
Factor Analysis of Solvent Shifts in Proton Magnetic Resonance by Paul H. Weiner, Edmund R. Malinowski,* and Alan R. Levinstone Department of Chemistry and Chemical Engineering, Stevens Institute of Technology, Hoboken, New Jersey (Receioed March 18, 1970)
070.90
Proton shifts of a series of simple substituted methanes are measured in a variety of solvents with TMS as an internal standard. A mathematical technique of factor analysis is developed and applied to the resulting data. This analysis indicates that only three factors are required to reproduce the data within experimental error. The solute shifts in three solvents (namely, acetonitrile, carbon tetrachloride, and methylene bromide) are chosen as test factors. This choice is not unique. Reasons for this choice are discussed. All solvent shiftsused in the scheme can be expressed in terms of these three factors. The method is successfully extended to solutes, such as benzene and acetone, which were not included in the original analysis. It is also shown that the gas-phase shift of a solute is indeed a factor. Where the data are lacking, gas-phase shifts are predicted. Other possible factors are also considered. Introduction The ultimate goal in the study of solvent shifts in nmr is to account for the behavior with a minimum set of variables. Present theories unfortunately are successful only in a qualitative or semiquantitative manner. The models upon which these theories are based are questionable because of the many crude approximations involved. In the present study the goal was to develop a mathematical technique, called factor analysis, * 4 in an attempt to decipher the number of controlling factors and to test the prevailing theories. The immediate aim was to develop a procedure for predicting the shifts of simple solutes in a large variety of solvents from a minimum of shift data.
Experimental Section The proton spectra were recorded with a Varian A60A spectrometer, operating at a probe temperature of
* T o whom correspondence should be addressed. (1) C . Spearman, Amer. J . Psychol., 15, 201 (1904); “The Abilities of M a n : T h e Nature and Measurements,” MacMillan, 1927; Brit. J . Med. Psychol., 17, 322 (1927); J . Educ. Psychol., 28, 629 (1937). (2) L. S. Thurston, “Primary Mental Abilities,” Chicago University
Press, 1938; “Multiple Factor Analysis,” Chicago University Press, 1947. (3) E. R. Malinowski, Ph.D. Thesis, Stevens Institute of Technology, 1961; Dissertations Abstract, 23(8) Abstract 62-2027 (1963). (4) P. T. Funke, E. R. Malinowski, D. E. Martire, and L. Z. Pollara, “Application of Factor Analysis t o the Prediction of Activity Coefficients of Non-electrolytes,” Separation Sci., 1, 661 (1966).
The J O U Tof ~Physical Chemistry, Vol. 74?N o . 86,1970
P. H. WEINER,E. R. MALINOWSKI, AND A. R. LEVINSTONE
4538
Table I: Chemical Shifts of Substituted Methane Solutes in Polar and Nonpolar Solvents, in Hz at 60 MHz, Relative to Internal TMS
CH4 CHaCN CHaCl CH2Clz CHCla CH3Br CHzBrz CHBra CHJ CHzIz CHI8 CHzClBr CHiClCN CHBrClz a
12.1 117.6 181.6 326.9 455.7 160.4 305.2 425.4 130.4’ 238.1 303.3 317.9 256.8 449.7
12.1 118.0 181.1 319.8 438.9 158.8 297.6 412.8 129.3 233.6 295.8 311.2 248,2 432.1
12.7 120.0 180.2 317.4 436.1 158.7 295.5 41O.Oa 129.6 232.1 294.5 309.4 246.1 430.4”
13.8 117.4 178.8 317.1 435.0 157.2 295.4 409.2 128.9 232.2 294.7 308.3 244.2 430.0
13.3 114.8 176.6 313.9 432.5 155.8 292.9 406.9 128.5 232.1 293.1 306.4 242.8 429.0
13.8 122.7 182.2 321.2 440.8 161.3 300.3 412.6 132.3 234.9 293.9 313.9 252.3 435.4
12.9 122.9“ 180.6 322.5 446.1 159.6 301.0 416.6 131.0 235.5 296.0 314.8 255.3 439.6
15.2 127.3 184.1 321.8 439.6 162.9 299.0 411.0 133.7 234.6 292.5 312.4 253.0 433.0
15.1 128.8 185.3 323.5 441.4 163.7 301.0 410.9 135.7 235.0 288.2 315.6 257.8 434.9
Deuterated solvent was employed.
39 f 1’. The spectra were calibrated using a HewlettPackard Model 200hB wide range oscillator and a Hewlett Packard Model 523DR frequency counter. All compounds were used as obtained from commercial sources. The solutions were prepared by pipetting 1 drop of solute into approximately 1 cc of solvent, which contained a trace amount of TMS as an internal standard. It was not necessary to degas (remove O2 from) the samples since an internal standard was used. Deuterated solvents were used when the solvent peak obscured the solute spectra. The chemical shifts obtained are shown in Table I. Mathematical Formalism of Factor Analysis. The key steps involved in factor analysis are shown in Figure 1. First a matrix of experimental data is converted into a correlation matrix. Linear factors which reproduce the original data are obtained from the correlation matrix. These factors can be mathematically rotated into physically significant parameters, which also account for the experimental data. Factor analysis is based upon expressing a property as a linear sum of terms, called factors. This analysis seems applicable to proton solvent shifts since almost all investigators believe that solvent shifts are a linear sum of contributions, namely, anisotropy, bulk magnetic susceptibility, van der Waals effects, reaction field, etc. I n this perspective we express the proton shift Sik of solute i in solvent k as a linear sum
EXPERIMENTAL
CORRELATION CORRELATION
CHEMICAL SHIFTS
MATRIX
[cl
Sik
REPRODUCTION
I
I
PHYSICALLY
LINEAR FACTORS
SI GNlFlC IENT
PARAMETERS
gij AND Vjk
I
Figure 1. Key steps in factor analysis.
Both the solute-factor matrix [ U ]and the solvent-factor matrix [VI can be constructed strictly from a knowlthe shift edge of the matrix of experimental values [SI, matrix. To achieve this a square symmetric correlation matrix, [C], of dimension r X T , is constructed by taking the product of the shift matrix, premultiplied by its transpose (3)
Matrix [ C ]can be diagonalized by a matrix [ B ]
-’[c1 lB 1 =
[B1
where U i j is a solute factor and v j k is a solvent factor, the sum being taken over r factors. Factor analysis is designed t o tell us how many factors are involved. In matrix form eq 1 becomes
[SI
=
iVl[Vl
The Journal of Physical Chemistry, Vol. 74,No. 96,1970
(2)
[x$jk
1
(4)
( 5 ) R. B. Catell, “Factor Analysis,” Harper and Row, New York, N. Y., 1952. (6) K. J. Holainger and H . H. Harman, “Factor Analysis,” Univ. of Chicago Press, Chicago, Ill., 1941. (7) D. N. Lawley and A. E. Maxwell, “Factor Analysis as a Statistical Method,” Butterworths, 1963.
FACTOR ANALYSIS OF SOLVENT SHIFTSIN PROTON MAGNETIC RESONANCE Here 8 j k is the Kronecker delta. A, is an eigenvalue of the set of equations
[CI{B,J
=
bbl
(5)
wherej = 1,2, ..., r and { B, f is the corresponding eigenvector. These eigenvectors are orthogonal and can be used as a basis set. Now
[B-’1 [C][ B ]= [B]-’[S]T[S] [ B ]= [BIT[fJIT[fJl[Bl= [UITIUl =
[V,fil
where [UI = [SIP1
(6)
Thus the shift matrix can be expressed in terms of [ B ] and [SI
[XI
=
[UI[BlT
(7)
Since eq 7 expresses the same relationship as eq 2, then
[BIT = [VI (8) The problem, however, is t o reproduce [SI within experimental error using the minimum number of linearly independent eigenvectors. As a first trial we start with the eigenvector { BI ] associated with the largest eigenvalue XI, and perform the following matrix multiplication
[SI= [Ull[B11
(9)
where u1 = { uil),a column vector, and B~ = { v l k ) , a row vector. We proceed by utilizing the next largest eigenvector, and the next one, and so forth until the shift is satisfactorily reproduced. The minimum number of eigenvectors required will exactly equal the dimensionality of the factor space; ix.,the number of factors involved, namely r. In other words
4539
effects. To effect such rotations we simply “guess” what these factors might be and then attempt to reproduce the data as indicated in Figure 1. Mathematically, rotation of the axes is accomplished as
[VI=
[UI[Rl (11) where [R] is the rotation matrix and [D] is the rotated solute-factor matrix. The inverse of the rotation matrix is used to locate the solvent-factor matrix in the new coordinate system, i.e.
[VI
=
[Rl-l[Vl
(12)
A least-squares method for obtaining the rotation matrix and for testing suspected parameters is readily deduced in the following manner. Consider a column of the rotation matrix, see eq 11. In the lcth column is located the vector (RIa,R~k,...,R,a), which when multiplied by the U,j components of the ith solute gives its position, in the new coordinate system. The difference between the rotated solute shift and the actual value 6,, is
uta,
= Utr,
-
UeRir,
ut,
=
Uta =
+ U&a + . . . + UijRja
-
Vir,
(13)
When this difference is minimized by standard leastsquares procedures, the following equation results. {R} = [YI-’{XJ where [ Y ] {, X
(14)
1, and { R ] are defined as
Rza =
At this point the factor analysis is essentially complete. The number of factors necessary t o account for the original data has been deduced. Referring to Figure 1 (see dotted line marked REPRODUCTION) the linear factors as expressed in eq 10 reproduce the experimental chemical shifts. In their present forms the solute and solvent factors are not recognizable in terms of physical or chemical quantities. Instead they merely represent mathematic solutions to data reproduction. From the viewpoint of a chemist it is desirable to rotate the mathematical reference axes into axes which have physical significance. This procedure would provide an insight into the true fundamental factors which are operative in solvent
: R/k
[YI
=
’
cu;1ui*
cuizuu. .Cr;l,Z ,
j
(17)
~
From eq 6, 16, and 17 we see that eq 14 can be rewritten as IRj
=
[(l/W,r,IWIT{i71
(W
1,
The least-squares, vector rotator { R a column of [ R ] , is readily calculated from eq 18. If our suspected paThe Journal of Physical Chemistry, Vol. 74, N o . 26, 1.970
P. H. WEINER,E. R. MALINOWSKI, AND A. R. LEVINSTONE
4540
1
1
rameters { are true factors then 1 must equal { ] within experimental error. The elements of are obtained from eq 11; namely
{u]
The least-squares method for rotation as developed here is completely general and is applicable even when some values are either unknown or purposely omitted. In this situation, of course, appropriate terms must be removed from the summations in eq 15 and 17. This procedure has a hidden advantage; it automatifor those cally yields, and thus predicts, a value of Vi, values were omitted. molecules whose DfJc The final step in factor analysis is simply to regenerand [PI, ate the shift matrix [SIusing the rotated i.e.
vi&
[a]
[SI= ID1 [PI
(20)
Factor Analysis of Solvent Shifts. Two criteria must be met in order to apply the technique of factor analysis to the problem of nmr solvent shifts. One is that it must be possible t o separate the solvent shift into a sum of linear terms. Second, each term must be a product function of a solute and solvent factor. This places severe restrictions on the types of data which can be factor analyzed. Buckingham, Schaefer, and SchneideP have postulated that the solvent shift can be expressed as a linear sum of terms. The chemical shift of a solute molecule i in a solvent k is given by the following equation Sik
= &(i) f ab(k)
f ga(k) f uw(i,k)f u&k)
(21)
where 6,(i) is the gas-phase shift of solute i, ab(k) is due to the bulk susceptibility of the solvent k; a,(k) is the solvent shift caused by the anisotropy of the solvent) k; uw(i,lc) is the van der Waals or dispersion interaction effect between the solute and the solvent; and uE(i,k) is the reaction field interaction between the solute and solvent. The individual expressions for the various terms can be expressed as a product function of solute and solvent parameters, under special circumstances. These circumstances are discussed in a later section. All factors need not be explicitly identified in order to use the technique of factor analysis. Factor analysis yields the minimum number of independent factors necessary to span the solvent-effect space. In many instances the exact number of factors may be somewhat indecisive due t o experimental error of the data points. In such instances the cutoff is usually taken when the data are reproduced within experimental error, and the introduction of another factor does not significantly improve the fit, A group of solvents can be judiciously chosen which, separately or in conjunction, contains all of the suspected solvent effects. As an example, if hydrogen bonding effects were involved, then at least one of the solvents chosen to span the factor space must exhibit this type of interaction. The chemical shifts of The Journal of Physical Chemistry, Val. 7 4 >
86, 2970
solute molecules in these solvents can then be used as test factors. We could then hopefully obtain equations which would predict the shifts of the solutes in other solvents from measurements in the test solvents. For the shift matrix, Table I was employed, purposely excluding CH,, CH3CN, CHZCL,CH2ClCN, and CHBrClz as solutes for later testing purposes. Subjecting this matrix of data to the factor analysis computer programg we obtained the following eigenvalues: X ( l ) = 9.0; x(2) = 2.4 x 10-4; ~ ( 3 )= 5.5 x 10-6; ~(4) = 8.0 X X(5) = 3.0 X X(6) = 2.0 X ~ ( 7 )= 8.0 x 10-7; ~ ( 8 = ) 4 x 10-7; ~ ( 9 )= 9.0 x Each eigenvalue is a measure of the relative importance of the corresponding eigenvector. By referring to the reproduction of the shift matrix with r factors (see discussion concerning eq 9 and lo), we find that only three factors are required. With three factors the average error for data reproduction is less than h0.5 Hz, which is well within experimental error. At this stage, in principle, the factor analysis is complete and the original data can be reproduced with the three fundamental factors (see dotted line in Figure 1). However, it is more convenient for us to express the shifts in terms of physically significant factors. For this reason we decided to rotate the eigenvectors into three solvent vectors (acetonitrile, carbon tetrachloride, and methylene bromide). These three solvents were chosen on the expectation that they adequately span the solvent space. Acetonitrile possesses a large dipole moment and has .R electrons. Methylene bromide has a large polarizability and a sizeable quadrupole moment. Carbon tetrachloride is nonpolar and contains bulky chlorine atoms. Although there is nothing unique about this choice, one must use caution since any three solvents Will not necessarily span the factor space. For example, methylene chloride, chloroform, and carbon tetrachloride jointly do not satisfactorily reproduce the data; evidently one factor is not sufficiently represented by this group. From the rotated solvent factor matrix [VI, we obtain a series of equations that predict the chemical shift of a solute in a given solvent from the measured shifts in the three chosen solvents. The resulting equations are
S~,CH*CN = l.OOl6fi - O.OO38f2
+ O.O022f,
+ 0.716Ofi + 0.207Of3 + O.8169fz + 0.23oOf3 X ~ , C H C ~ -0.0458fi ~ Xi,cclr -O.OOlSf, + 1.0041f2 - O.oo22f3 Si,cs2= 0.0064fi + 1.1281fi - 0.1394.fa XI,CH2cll= 0.0806fi
=:
(8) -4.D. Buckingham, T. Schaefer, and W. G. Schneider, J. Chem. Phys., 32, 1227 (1960). (9) A computer nrogram and listing is available on request from the authors.
4541
FACTOR ANALYSISOF SOLVENT SHIFTSIN PROTON MAGNETIC RESONANCE Table I1 : Comparison of Calculated and Experimental Chemical Shifts of Substituted Methanes5 C -H4Exptl
Solvents
12.1 12.1 12.7 13.8 13.3 13.8 15.2 12.9 15.1 3.0
CHaCN CHzClz CHCla cc14
cs2
CH2Brz CHBra CHJ CH212 Exptl range of data Av error 5
---
Solutes
7 -
Pred
13.7 13.8 13.6 14.3 12.7 13.8
,--CHaCNExptl
Pred
117.6 118.0 120.0 117.4 114.8 122.7 127.3 122.9 128.8 14.0
119.4 118.8 116.1 124.6 119.8 128.1
-CHaCICNExptl
7
Pred
256.8 248.2 246.1 244.2 242.8 252.3 253.0 255.3 257.3 15.0
247.5 245.7 241.9 252.3 254.2 258.5
1.7
0.9
Y C H z C l m Exptl Pred
326.9 319.8 317.4 317.1 313.9 321.2 321.8 322.5 323.5 13.0
319.5 317.9 315.0 320.9 322.2 321.7
----CHClBr2--7 Exptl
449.7 434.5 432.1 430.2 427.9 435.4 433.0 439.6 434.9 21.8
434.0 431.0 427.5 433.3 440.4 434.9 0.5
0.8
0.7
Pred
These solutes were not included in the original factor analysis scheme.
~
Table 111: Comparison of Experimental and Predicted Chemical Shifts of Various Solutes5 Using the Solute Chemical Shifts in CHaCN, CH2Br2,and CC14 as Solvent Factors
-
Solvents
CHsCN CClr CHzBrz CHCls
csz
CHaI Exptl range of data Av error of predictions a
U".U"VY
,--CHsClaExptl
Pred
-CH&lCClsExptl Pred
165.2 269.0 163.3 255.8 165.8 261.8 163.3 164.0 257.2 256.8 161.9 162.1 254.1 253.8 165.7 164.7 263.0 264.6 3.8 14.9
FCHC12CClaExptl Pred
395.4 361.7 363.0 347.3 372.7 354.0 366.3 364.1 350.6 348.6 362.1 360.0 345.1 344.8 377.2 384.2 354.4 356.4 33.3 16.5
0.8
0.6
-CHsCHBrExptl Pred
4.0
1.6
-(CHs)zCHBrExptl Pred
.--AcetoneExptl Pred
262.3 124.5 252.0 125.4 258.7 128.8 267.3 253.3 129.7 126.4 248.8 249.8 122.4 124.3 256.7 259.7 126.1 125.9 13.5 7.3 2.7
1.3
--Benzene Exptl Pred
442,7 435.9 441.2 441.1 437.3 433.4 433.0 439.1 438.9 9.3 1.4
These solutes were not included in the original factor analysis scheme.
Xi,CHzBr#
Xi,CHBra
+
+ O.9922f3 = -0.2562f1 - 0.0527fi + 1.3118fa O.5613f1 - 0.2237fZ + O.6527f3 = -0.1092f1 - 1.1972f2 + 2.2946f3 = -0~oO~of~ 0.0090f~
Si,CHaI Si,CHzI*
where fl
=
Xi,CHaCN) fi
=
Si,CClr,
and f3 =
Si,CHrBra
The equations above for acetonitrile, carbon tetrachloride, and methylene bromide, the test factors, each exhibit three finite coefficients. This is due to experimental error and computer roundoff. The accuracy of these equations can be tested on the solutes which were purposely left out of the factor analysis scheme. The results are presented in Table 11. For the simple substituted methanes, the agreement, although slightly beyond experimental error, is reasonably satisfactory. As a test of the generality of these equations we have calculated the chemical shifts of some nonmethane so-
lutes. These data are presented in Table 111. The agreement, especially for such solutes as benzene and acetone, is quite surprising. A Search for the Three Fundamental Factors. As described in the previous section, factor analysis shows that three factors span the solvent-effect space. These factors were rotated into three solvent vectors (acetonitrile, carbon tetrachloride, and methylene bromide). Evidently the three fundamental factors are sufficiently contained within these three solvents but have not been identified by this procedure. I n principle we should be able to rotate the factors resulting from factor analysis into the true fundamental factors. According to Buckingham-Schaefer-Schneider,8 see eq 21, the gas-phase shift of the solute should be a fundamental factor, since these shifts represent those of the unperturbed solute molecules. I n this case the solute factor U t j = (gas) and the solvent factor V j k = 1. Rotation into the gas-phase shifts was indeed successful as shown in Table IV. The agreement between the experimental values and those resulting from factor analThe Journal of Physical Chemistry, Vol. 7 4 , N o . $6, 1970
P. H. WEINER,E. R. MALINOWSKI, AND A. R. LEVINSTONE
4542
Table IV: Test' of Gas-Phase Chemical Shifts as a Solute Factor Using Three Factors in the Rotation Matrix" 8,
% ,
SOhtE
(predioted)
(expt1)O
CHIC1 CHCla CHsBr CHzBr2 CHBra CHaI CHzIz CHI3 CHzClBr
168.2 427.1 147.1 286.5 406.8 118.5 227.6 301.5 297.7
Difference
...
...
427.3 146.9 285.0 406.9 119.0
-0.2 0.2 0.5 -0.1 -0.5
... .,.
...
... ...
...
I n hertz at 60 MHz, relative to gaseous TMS. 'These values ' The values correspond to Ti. correspond to
vi.
ysis is well within experimental error. One bonus gained by identifying the gas-phase shift as a solute factor is that gas-phase shifts are automatically predicted for the molecules where data are not available. The predicted values are also shown in Table IV. The reaction field term, as developed by Buckingham, Schaefer, and Schneider18under appropriate conditions, has the form
where subscripts u and v again refer to solute and solvent; p is the permanent dipole moment; a is the polarizability; 8 is the angle between p and the C H bond of the solute proton in question; n is the index of refraction; x is a constant for the C H bond; B is the dielectric constant. Equation 22 can be factored into a solute and solvent term if an average value of nu2is substituted into this relation; namely, nu2F= 2.5. Thus
The Journal of Physical Chemistry, Val, 74, No. 16,1070
I n this form the solute factor (yu/au cos 6 ) can be tested by the procedure described previously. Surprisingly, rotation into this suspected fundamental factor was unsuccessful. There are several possible reasons for this failure. The total reaction field effect may be so small that it is obliterated by experimental error. Furthermore, quadrupolar effects may not be negligible, as is commonly assumed. The van der Waals factor, uw, can be expressed as a product function for nonpolar solutes in nonpolar solvents.'0-12 However, for polar solutes in polar solvents, theoretical formulas have not been developed. Consequently, no formula exists for testing by factor analysis. The anisotropy shift is a product function in which the solute factor Uij = 1 and the solvent fact V,, = u, (solvent). Since the present study involves internal standards this effect cannot be one of the three fundamental factors.
Acknowledgment. The computations were carried out at the Computer Center (supported in part by a grant from the National Science Foundation) of Stevens Institute of Technology, for which we record our appreciation. Also, the authors gratefully acknowledge the support of the U. S.Army Research Office (Durham), Contract No. DA-31-124-ARO(D)-90. (10) A. A . Bothner-By, J. Mol. Spectrosc., 5 , 52 (1960). (11) H. J. Bernstein, private communication. (12) E. R. Malinowski and P. H. Weiner, J. Amer. Chem. SOC.,92, 4193 (1970).
