EILEEN MCLAUGHLIN AND R. W. ROZETT
1860 an order of magnitude for rR 2 1 X lovBsec. We find that linear interpolations along the vertical line between the curves A and B (or C and D, or E and F) give the correct results for intermediate line width values. The curves in Figure 1 were calculated for isotropic rotational reorientation. For anisotropic diffusion, the approximation of axial symmetry for the spin parameters may no longer be valid and the anisotropic parameters must be used. For simplicity, the rotational diffusion tensor can often be assumed to be axially symmetric Kith its symmetry axis x’ = x, y, or x of the molecular fixed axis.Io Thus RII and RI are the components of the rotational diffusion tensor about the x’ and the x’ and y’ axes, respectively. For x’ = Y and RI1 > RI, i.e., fast rotation about the molecular z axis, the results are relatively straightforward. This type of rotation preserves the approximate axial symmetry of the spin parameters, and the observed value of S is the value expected for isotropic diffusion and T R = (6R1)-1. For relatively more rapid diffusion about the x or y axes, the results are more complicated. For small anisotropies about these axes, i.e., R I I 5
3RL, the value of S is very slightly changed from the value calculated for isotropic diffusion and T T ~= ) (SR)-l = l/6(RIRl,)1’*,but this corresponds to a decrease of about 8% in the apparent value of rR obtained from Figure 1. For larger anisotropies, a decrease in the value of S is observed (e.g., for rfi = 3.0 X 10-8, Brownian diffusion, and a line width = 0.3 G, S decrease from 0.931 for Ri 1/RI = l to 0.897 for R II/R, = 20 or an apparent decrease in T R obtained from Figure 1 by a factor of 2). The magnitude of this decrease is independent of whether the x or y axis is the symmetry axis. However, in general, if the axis of rotation is unknown, or does not correspond to a molecular coordinate axis, or if the rotation is completely asymmetric, then estimates of the components of the rotational diffusion tensor can only be obtained from detailed spectral simulations.8 Further detailed studies of other nitroxides in terms of the general theory4J and the simplified method discussed here are of interest and are being pursued in part in our laboratories. (10) This is a good approximation for R.‘ defined as Rl = (Rz’ 4-R,’)/2.
>>
Rz’,R,’ with Rl
The Monoisotopic Mass Spectra of Borane Derivatives1 by Eileen McLaughlin and R. W. Rozett * Chemistry Department, Fordham University, New York, N e w York
10468
(Received November 9, 1971)
Publication costs assisted by the Petroleum Research Fund
The monoisotopic mass spectra of borane derivatives are calculated by a least-squares computer technique from polyisotopic information. Alkyl boranes, carboranes, and borane derivatives containing bromine, chlorine, sulfur, nitrogen, oxygen, deuterium, many metals and any monoisotopic element can be handled. Elemental formulas for ions can be established. As examples we investigate the following borane derivatives : &,JH~N~, C Z B ~ HCH~CBSHE, ~, (CH&C2BeH8, (CH&CzB7H,, and (c&)&~&Hs. Finally the sources of error in the procedure are discussed.
Introduction The mass spectra of borane derivatives which contain elements with a significant fraction of a second or third isotope are even more complex than the polyisotopic mass spectra of the boranes, B,H,. For example, in the mass spectrum of decaborane-14, in the mass region of ten boron ions, 165 isotopic variants occur if one takes into account only the boron isotope combinations. If in addition the deuterium variants are counted, 1254 ions could be found. The small natural abundance of deuterium makes the hydrogen case trivial, but the same cannot be said for the presThe Journal of Physical Chemistry, Vol. 76, No. 18,1.979
ence of in alkyl boranes and carboranes, or for 37Cland *IBrin haloboranes (see Table IX). If the mass spectrum is taken to study kinetic or mechanistic details of reactions, the complicated polyisotopic data may be looked upon as a mask hiding monoisotopic information. But from another point of view, the isotopic variation contains additional information which may be used to identify the ions present (1) This research was supported in part by the Petroleum Research Fund administered by the American Chemical Society (PRF 1233G2). The instrumentation was supplied in part by the National Science Foundation (GP 8220) and by the New York State Science and Technology Foundation (NYSSF (6)-13).
THEMONOISOTOPIC MASSSPECTRA OF BORANE DERIVATIVES
1861
Table I : Isotopic Variants of CZBZ+ r(W) 44 45 45 46 46 46 47 47 48
2! 2 I.
i! 1. 0 1. 0 0
0 0 1 0 1
2 1 2 0 1 2 0 1
2
1 2 2
0
r(n,m)
1.0000 1.0000 2.2449 X 10-2 1.0000 2.2449 X 1.2599 X 2.2449 x 10-2 1.2599 x 10-4 1.2599 X
in the spectrum. It is hardly novel to use the characteristic cluster of intensities due t o isotopic variants to determine the elemental composition of ions.2 However, the technique is especially useful in the boranes because of the significant percentage of loB (20%). Ditter, Gerhart, and Williams showed how the formulas of carboranes could be inferred from their characteristic isotopic pattern^.^ Their technique used successive approximations and was a tribute to their persistence. I n this paper we report a computer method which automatically and exactly determines the elemental composition of borane ions and the ions of borane derivatives from their polyisotopic mass spectra. The computer program (Table 11) calculates the exact isotopic cluster for boron and another element with a pair of isotopes and provides a least-squares-fitted monoisotopic spectrum. While calculating the monoisotopic spectrum a quantitative measure of the fit of the result is generated. Using this, one may choose between alternate elemental compositions for all the ions in the spectrum. I n this respect the procedure represents a substitute for the high-resolution mass spectroscopy of the ~ a r b o r a n e s . ~ After discussing the theory of the clusters of intensities due to several elements with isotopic variants, we illustrate the technique with BlOHBNl and some alkyl boranes and carboranes. Other borane derivatives which may be handled in this fashion are also discussed. Finally, the sources of error in the procedure and its limitations are detailed.
Method The procedure for calculating the monoisotopic mass spectra of molecules containing several elements, each with a pair of isotopes, can be formulated as follows. The measured intensity at any mass, Pt, is the sum of the abundances, ptj, due t o the different formulas which have one or more isotopic variants at that mass. These in turn may be expressed in terms of the intensity of the monoisotopic ion with the same formula, m,, as in pi =
CP,, = t=rirm., j 3
(1)
To calculate a monoisotopic mass spectrum one must
5.9505 x 4.8787 x 5.9505 x 1.0000 4.8787 x 5.9505 x 1.0000 4.8787 x 1.0000
r(k,l;n,m)
10-1 10-2 10-1
10-1
5.9505 X 4.8787 X 1.3358 X 1.0000 1.0952 x 7.4970 X 2.2449 x 6.1467 X 1.2599 x
rij
5.9505
x
10-2
10-1 4.8921 X 10-1 1.0110 10-6 10-2
2.2511 x 10-2 1.2599 X
10-4
first evaluate the coefficients, r t l r from statistical considerations. Then the simultaneous linear equations described by (l), one equation for each experimental intensity, must be solved for the m,. The r41 may be calculated from the following eq 2-6.6 The procedure is illustrated for the formula GB2+ in Table I. We assume that the ion under discussion has n atoms of one isotope of an element and m atoms of the second isotope. The fractional abundance of the ion, u(n,m), is a function of the gross abundance of the first isotope, f1, and of the second isotope, f2 a(n,m) = w(n,m)flnf2"
(2)
The statistical weight factor, w(n,m), is a binomial coefficient. It expresses the number of ways of arranging n things of one kind and m things of another kind without regard to order w(n,m) = (n
+ m)!/n!m!
(3)
The relative abundance of the ion, r(n,m), is the ratio of the fractional abundance of the ion to the abundance of the pure isotopic variant of the same formula
+ m)
r(n,m) = a(n,m)/a(O,n
(4)
For example, in Table I the abundance of log2+ relative This to I1B,+ is listed under r(n,m) as 5.9505 X number was calculated by inserting n, Le., 2, the number of atoms of the lighter isotope, m, i.e., 0, the number of atoms of the heavier isotope, and the natural abundance of the boron isotopes into eq 2, 3, and 4. Let us now suppose that the ion in question contains two elements, each with a pair of isotopes and that there are n and m atoms of the isotopes of the first element, and k and E atoms of the two isotopes of the second element, respectively. The abundance of the (2) J. H. Beynon, "-Mass Spectrometry and Its Applications t o Organic Chemistry," Elsevier, Amsterdam, 1960, p 305. (3) J. F. Ditter, F. J. Gerhart, and R. E. Williams, Advan. Chem. Ser., No. 72, 191 (1968). (4) R. L. Middaugh, M. T. Brady, and W. L. Budde, Midwest Regional Meeting of the American Chemical Society, St. Louis, Mo., Oct 1971. (5) B. Parl, "Basic Statistics," Part 111, Doubleday, New York, N. Y . , 1967.
The Journal of Physical Chemistry, Vol. 76, N o . 19,1978
EILEEN MCLAUGHLIN AND R. W. ROZETT
1862
~
Table I1 C C C C C C
C C C C C C C C C C C C C
C C C
C C C
MICS 1 MICS 2 3 MICS 4 R .W R O Z E T T C H E M I S T RY 0 EP AR TM E N T 5/71 MXCS 5 FORDHAM U N I V E R S I T Y BRONX 9 N Y 1045 8 MICS b MICS D I M E N S I O N A ~ 1 0 0 0 l ~ P K 1 2 5 0 ~ ~ P O L M A S ~ l 2 5 ) ~ P K S U M ~ l 2 5 ~ ~ N AM MI CES~ l O7~ ~ 1 A U X ~ 2 0 0 ~ ~ I P I V 1 1 0 0 ~ ~ I B ~ l O ~ ~ ~ I H ~ l O ~ ~ ~ I C ~ l O ~ M l I~C IS X ~ 8l O O ~ ~ O N E 9 2PKONE(100) MICS DATA IDNO/O/ ~ E P S / 1 ~ O E - 0 7 / t F R C T B 1 / o 8 0 3 9 / ~ F R C T 8 0 / ~ 1 9 6 1 / M I C S10 D A T A MAX1/100O/rMAX2/258/rHAX3/125/ *MAX4/lOO/ M X C S 11 M I C s 12 A L T E R N A T E I S O T O P E S OF B 1 0 / 8 1 1 D E R I V A T I V E S M I C s 13 H H/D C12/C13 BR79/81 CL35137 N14/N15 M I C S 14 014/018 S32/S34 MICS 15 M I C S 16 F R A C T I O N A L A B U N D A N C E AND M A S S E S OF O T H E R E L E M E N T S M I C S 17 DATA FRCTCZ/Q~000Q/~fRCTC3/loOOOO/~MASCl3/OO/vI~If/l/ M I C S 18 D A T A F R C T C ~ J . ~ ~ ~ ~ ~ / , F R C T C ~ / O O O O ~ ~ / ~ M A S C ~ ~ / O ~ / ~ ~ MDI C I SF / 1~9/ D A T A FRCTC2/0o9889/rfRCTC3/OoOlll/~MASCl3/l3/~IDIF/l/ MSCS 2c DATA FRCTC2/0.5054/rFRCTC3/0.4946/1MASC13/8l/,IDIF/2/ MICS 21 DATA f R C T C 2 / 0 . 7 5 5 3 / r F R C T C 3 / O o 2 ~ 4 7 / , ~ A S C l 3 / 3 7 / , I D I ~ / 2 / MJCS 2 2 DATA FRCTC2/009953/~FRCTC3/0.0037/rMASCl3/15/tIDiF/l/ M I C S 23 DATA FRCTC2/~99759/,FRCTC3/~~0204/~MASC13/18/~IDIF/2/ M I C S 24 DATA FRCTC2/Jo950Q/rfRCTC3/0.0422/rMASCr3/34/,IDIF/2/ MICS 25 M I C S 26 T I T L E C A R D M I N / M A X M A S S I N P O L Y S P E C T R A s M I N / M A X NO OF BORQNS. M I C S 27 H Y O R O G E N S AND O T H E R E L E M E N T S M I C S 28 114 R E A D ( l r 4 , f N O = 9 9 ) J C O D E , M A S S M N , M A S S M X , N A M E , N U M ~ M X , N U ~ H M X , ~ U M C M X , ~ U M 3 M ~ C S 29 1 M N r NUMHMN, NUMCHN M I C S 30 4 F O R M A T ( 31 5 9 1 O A 4 , 6 I 2 r 5 x 9 4 1 2 ) MlCS 31 95 N U M P O L - M A S S M X - M A S S M N + l MICS 32 20 ADD=O.O M I C S 33 ICOL-0 M I C S 34 MAXROW=NUMPOL M I C S 35 36 11 00 7 9 I = l , M A X l MICS A ( I)=O.O M I C S 37 79 C O N T I N U E M I C S 38 DO 33 I t l i M A X 4 M I C S 39 O N E M A S ( I ) = 0. H I C S 40 PKONE(S)=O. MICS 41 P K S U M ( I)=O. M I C S 42 33 CONTINUE M I C S 43 DO 3 2 I = l r M A X 2 M I C S 44 PK( Ir = f l o O M I C S 45 32 C O N T I N U E M I C S 46 M I C S 47 DATA I N P U T M I C S 48 I N T R O D U C E T H E I N T E N S I T I E S F R C M LOW M A S S TO H I G H M I C S 49 98 R E A D I 1 9 3 ) f P K S U M I I ) , I = l i N U M P O L ) M I C S 50 M O N O I S O T O P I C B O R A N E MASS S P E C T R A BORON-CARBON-HYDROGEN MASS SPECTRA
? FORMAT(16I5) 3 FORMAT I12F6.21 C C C C
I N T R O D U C E T H E F n R N U i A S FROM LOW M A S S ' T O H I G H M A S S B L A N K CARD AFTER L A S T FORMULA NUMX IS T H E M A S S A P A R T FROM C v H v B 651 READ(~~Z~END=~~JNUMB(NUMH,NUMH~NUMC~NUMX IF(NUNB+NUMH+NUWC*NUMX,EQ.O) GO T O 655 652 M A S S 2 = N U M B * l l + NUMH+NUMC*MASC13+NUYX lF(MASS2.iToMASSMN) GO TO 6 5 1
MASSl=MASS2-NUMB-NUMC*IQIF
The Journal of Physical Chemistry,Vol. 76,No. IS,1972
MICS MICS MICS MICS MICS MICS MICS MICS MICS MICS MICS
51 52 53 54 55 56
57 58 59
60 61
1863
THEMONOISOTOPIC MASSSPECTRA OF BORANE DERIVATIVES
Table I1 (Continued) MICS MICS MICS
ICOL=f COL+1
ONEMAStICOL)=MASS2-NUMC*IDIF IB(ICOL)=NUMB IH ( I C O L 3 =NUMH I C ( I C O L )=NUMC I X ( ICOLI=NUMX N11M6 X = N UH B+ 1 NU =HASSl-MASSMN+4ICOL-l~*MAXROW NUMCX-NUMCtl IF'(MASSl.GEoMASSMN9GO T O 660 IO(ICOL)=999 C
HICS MICS MICS MICS MICS MICS MICS MI C s MICS MICS MICS
C O E F F I C I E N T M A T R I X OF S I M U L T A N E O U S E Q N S 660 D O 6 5 4 N U M B l l = l r N U M B X NUM=NU+NUMB 11 BABUN=RLABUN(NUM6~NUMBllrFRCTBlrFR&TBO) DO 659 N U M C 1 3 = l r N U M C X NUWClZ=NUMC-NUMC13+2 IF(IB(ICOL).NEo999)GO T O 657
MICS
MAS=MASSl+NUMC13*IDXF+NUMt311-IDIF-l 1FlMAS.LToMASSMN) GO T O 6 5 8 ABNUMl=A(NUM)+RLABUNI"C1YUMC12,FRCTC21F~CTC3)~6ABUN NlJM=NUM+IDIF CONTINUE CflNJXNUE GI3 T O 651 655 M A X C O L = I C O L
657 658 659 654
C
C C
C C C C
7 8 r = I , NUMPQL POLMAS(I)=MASSMN+4DD ADD=ADD+lo 78 C O N T I N U E N= MUMPOL + 1 011 200 I = N p M A X 3 POLMAS(I)=O. 200 C O N T I N U E
on
DATA PRINTOUT WRITE(3r341NUMBMX,NUM"UMC#X,N~MCMX,~~CTBl,NUMBMN,NUM~MN,
67 68 69 70 71
72 73 74
75 76
M I C S 77 M I C S 78 M T C S 79 M I C S 80 M I C S 81 M I C S 82 M I C S 83 M I C S 84 MICS 8 5 M I C S 86 M I C S 87 M ICS 88 M I C S 89 M I C S 90 M I C S 91 M I C S 92 MXCS 93 MICS 94 M I C S 95 M I C S 96 M I C S 97 M I C S 98 M I L S 99 N U M C M N M I C S 100
1t F R C T C 2 r N A M E MICS 3 4 FORHAT('l',T20~'MONOISOTCI?IC MASS S P E C T R A OF B O R A N E S E D E R I V A T I V E S M I C S l'///' F R O M BORANE H'rI2r'H',f2~'C't12rlOX~'FRACTION B11= M I C S 2'rF6.4/' TO BORANE 8'r12r'H't12r'C',12,10X~mF~A~TION ClYICS 32= ' r F 6 . 4 , 1 5 X i l C A 4 / / 1 MICS W R I T E ( 3 , 780) MAXROW, M A X C O L MICS 780 F O R M A T ( ' 0 ' 9 7 2 0 9 ' M A X R O W = ' r I 5 p MAXCOL='tIS) MICS W R I T E ( 3 t 981 1 MICS WRITE(3r9HO)(POLMAS1I),PKSUM(I)rIrl,MAXROW~ MIC5 980 F O R M A T ~ 4 ~ 5 X ~ F l Q o 1 ~ F 1 0 . 4 1 ) MICS 981 F O R M A T ( ' 0 E X P E R I M E N T A L I N T E N S I T I E S AND MASSES '//I MICS MICS L E A S T SQ S O L U T I O N O F S I M U L T A N E O U S EQNS ( I G M S S P ) MICS CALL L L S Q I A t PKSUMI MAXROW, M A X C O L r LrPKONE, I P I V , EPSI I E H i M I C S 1 AUK) MICS MICS F R A C T I O N OF SUM OF P E A K S A N D B A S E R E L A T I V E I N T E N S I T Y MICS SUMONE= 0. MICS ONEMAX= PKONE( 1 $ MICS DO 3 9 I = l r M A X C O L MICS IF(PKONF(I))79,39,40 MICS 40 SUMUNE= SUMONE + PKONEIII MICS I F (PKQNEtI) IINEMAX) 3 9 , 39, 3 8 MICS
-
62 63 64 65 66
101 102 103 1C4
195 106
107 108 109 11C 111 112 113 114 115 116 117 118 119 120 121 122 123
The Journal of Physical Chemistry, Vol. 7 6 , N o . IS, 1972
EILEEN MCLAUGHLIN AND R. W. ROZETT
1864
Table I1 (Continued) 3 8 ONEMAXPKONECI) 39 C O N T I N U E ONEMAX= loo. /ONEMAX SUMONE- 100./SUMONE
C C
MICS MICS MICS MICS MICS
SOLUTION PRINTOUT MICS WRTTE13r35) MICS SUMNEG=O*O MICS 00 3 7 I = l , M A X C O L MICS CLAST= PKONE(I1*SUMQNE MICS IF( C L A S T )115r116,116 MICS 1 1 5 SUMNEG=SUMNEG+CLAST MICS 116 C O N T I N U E MICS RE L= P K O N E ( I 1 *ONEMAX MICS J=ONEMAS(I) MICS P K ( J )/=REI. MICS IF(PEL.LE*UmO) PK(J)-0.0 MICS I T O N E M A S ( 1 ) ,PKQNE( I ) ~ R E L ~ C L A S TIB( I I 1 r IH( I I r I C ( 11, I X ( I I M I C S WRITE13,36) 36 FORMAT1110r2F15.3rlOX~2Fl5~3~1(7X~4I51 MICS 37 C O N T I N U E MICS 35 F O R M A T t ' O P E A K NO. 3 1 1 MASS 811 I N T E N S I T Y MICS 2 REL FRCT B H C X '//I MICS 8 =MA XROW MICS AUX(1)= SQRT(AUX( 1)JB) MICS WRITE~3~117)IER~AUXIl)rSUMNEG YICS 117 F O R M A T [ '0 IER= *,t5, * L S T sa RMS = *.~15.7** NEGATIVMICS lES= 'rF16.5) MICS
IF4IDNO.EP.O) GO T O 114 L-ONEMASIl) LL=DNEMASIMAXCOL) W R I T E I Z T ~ )( P K ( I ) r I = L r L l ) GO T O 114 99 C A L L E X I T END C
C
68
61 62 60 64
65 63 66 67
124 125 126 127 128 129 130 131 132 133 134 135 136
137 130 139 140 141 142 143 144 145 146 147 148 149
150
MICS 1 5 1 HICS 1 5 2 M I C S 153
MICS 154 MICS 1 5 5 M I C S 156 M I C S 157 M I C S 158 MICS 1 5 9 FUNCTION RLAHUN(NrK,FRCTKl p FRCTK2) M I C S 160 REAL*8 FRCTK~FRCTL~UNUMIONCIM~~DNOMZTW M I C S 161 C A L C U L A T E S R E L A T I V E A B U N D A N C E S OF M I X E D I S O T O P E M O L E C U L E S M I t S 162 IF(N.NEsO)GO TO 6 8 MICS 163 RLABUN=1.0 M I C S 164 RETURN H I C S 165 FRCTK=FRCTKl MICS 1 6 6 FRCTL=FRCTK2 MICS 167 UNUM*l.OD 00 HICS 1 0 8 ONQMl=l.QD OD MICS 169 0NOM 2= 1..OD .60 M I C S 170 KUsK-1 M I C S 171 IF (KKi60r60161 M I C S 172 DO 62 I=l,KK MICS 1 7 3 DNOMl=DNOMl*DFLOAT(I) M I C S 174 UKK=N-K+l M I C S 175 iF t K K K 1 6 3 r 6 3 r 6 4 M I C S 176 DO 155 I = l , K K K M I C S 177 DNOM2=aNOMZ*nFLOAT(I) MICS 178 DO 6 6 I = l r N M I C S 179 UNUM=UNUM*DFLOAT(I) MIGS 180 W=UNUM/(ONOMl*DNOM2) R L A B U N I W ~ F R C T L * * ( ~ - K + ~ ~ / F R C T L ~ ~ ~ C ~ K * ~ K / ( ~ R C T K ~ ~ ~ N + ~ M) I~C S 1 8 1 M ~ C S1 8 2 IF ( R L A B U N . L T s 1.00-12 ) R L A B U N = l . O D - l Z MICS 183 RETURN M I C S 184 END
The Journal of Physical Chemistry, Vol. 76,N o . 13, 1978
THEMONOISOTOPIC MASSSPECTRA OF BORANE DERIVATIVES ion relative to that of the pure isotopic ion is the product of the separate relative abundances for each element. The probability of the occurrence of the isotopic variation in element 1 is independent of the isotopic arrangement in element 2
r(n,m;k,Z) = ?+L,m)r(k,l)
(5)
For example, in Table I the relative abundance of lZCl3Cl0B2+,~(1,1;2,0),is listed as 1.3358 X It is the product of the abundance of 12C13C relative to W2,2.2449 X lo+, and the abundance of logzrelative to "Bz, 5.9505 X If two or more ions with the same formula but with diff erent numbers of each isotope (n',m',k',Z') occur a t the same mass, the abundance of the ions in question relative to the pure isotopic ion is the sum of the separate relative abundances. The two probabilities are mutually exclusive r(n,m;Jc,Z;n',m';k',Z') = r(n,m;k,l)
+ r(n',m';k',Z')
1865
Table 111 : B I o H ~ N ~ Mass
Intensity
105 106 107 110 111 112 113 114 115 116 117 118 14,s 146 173 174
1.4 2.9 2.1 42.3 17.4 48.1 7.0 40.6 15.5 100.0 9.1 99.4 6.2 80.9 4.5 69.5 0.21
(6)
I n Table I for instance two isotopic variants of the formula CzBz+ occur at mass 45, 12C210B11B+, and 12C13C10B2+. Their relative abundances are 4.8757 X 10-l and 1.3358 x 10-3, respectively. The relative abundance of all isotopic variants of formula GBz+ which occur at mass 45 is the sum of these abundances, or 4.8921 X 10-l. With these formulas all the coefficients r d l may be calculated. The i index specifies the mass of the isotopic variant; the ,j indicates a specific elemental formula. Finally one must solve the simultaneous equations (one for each experimental intensity) for the monoisotopic intensities (one for each monoisotopic formula). A least-squares technique which avoids the pitfalls of the usual procedure for calculating monoisotopic mass spectra has been described in recent compilation of the monoisotopic mass spectra of the boranese6 The program of Table 11, R!tICS, is a generalization of the program published previously, hlIBS, and replaces it.
Examples Tables I11 to VI11 contain the results for six compounds derived by the MICS program from published polyisotopic mass spectra. I n Table 111, for example, the formulas and monoisotopic intensities for BIOHBNl are given.7 'These were derived in the following may. The published polyisotopic mass spectrum, the natural abundance of the boron and nitrogen isotopes, and a chemically reasonable set of formulas were fed into MICS. The program calculates a least-squares-fit set of monoisotopic intensities and provides the rootmean-square deviation of the derived monoisotopic intensities from the original data. Should negative intensities result, the solution is easily restricted to positive monoisotopic intensities by removing the formula of the offensive peak and resolving for the intensities.
Table 1V : C Z B ~ H B Mass
Intensity
69 70 71 72 73 74 75 76
20.5 9.4 20.3 8.8 100.0 70.4 2.2 57.7 0.49
Table V :
CHaCBbHs
Mass
Intensity
79 80 81 83 84 85 86 87 88 89 90
5.9 1.o 14.1 9.3 26.0 15.9 65.4 61.9 87.4 8.4 100.0 0.30
Other reasonable sets of formulas were then run, and the set with the lowest root-mean-square deviation is reported in Table 111. A natural abundance of 19.61y0 was used for loB and 0.37Oj, for 16N. Either of these figures could cede to new evidence. For boron es(6) E. McLaughlin, T. E. Ong, and R. TV. Rozett, J . Phys. Chem., 75, 3106 (1971).
(7) R . L. Middaugh, Inorg. Chem., 7, 1011 (1968). The Journal of Physical Chemistry, Vol. 76, N o . 15, 197d
EILEEN MCLAUGHLIN AND R. W. ROZETT
1866
Table VI : (CH&CJ3& Mass
Formula
Intensity
117 118 119 120 121 122 123 124 125 126
C&H3+ C4BaHa+ CaBeHs' CaBeHs' CdBi"+ C&sHs+ C4BeHsf CaBeHio' CaBeHii CiBeHiz' RMD
10.0 6.4 9.6 8.4 8.0 8.3 13.7 25.3 8.7 100.0 0.78
+
Table VI1 : (CH~)ZCZB~H.I Mass
Other Derivatives of the Boranes Table IX lists the more common polyisotopic elements which can be handled by the MICS program in addition to boron. As we note, only two isotopes of the elements are permitted, so sulfur and oxygen derivatives of the boranes are handled by neglecting I7O, *%, and 36S. These isotopes are present in such small abundance that the approximation is excellent. Any one of the following metals may replace any one of the elements in Table VIII: Cu, Ga, Rb, Ag, Sb, Eu, Lu, Re, or Ir. The common factor is the presence of a pair of isotopes (or a close approximation), with the lower isotope in sufficient abundance. The program solves for the monoisotopic abundance of the lighter isotope.
Table I X : Fractional Abundances in Borane Derivatives
Formula
128 129 130 131 132 133 134 135 136 137 138
Intensity
7 9Br
0.0 2.3 1.8 7.8 0.0 15.7 13.9 19.0 18.8 1.7 100.0 0.85
asci
Table VI11 : (CH3)2C~BsHs Mass
Formula
Intensity
140 142 143 144 145 147 148 149 150
C4BaHa C4BsHa C4BsHrf C4B& C4BeHs CaBsHii C&Hn+ CaBsHrs CiBsHia RMD
1.9 6.7 3.8 2.9 13.8 26.5 43.7 15.2 100.0 0.94
+
+
' +
+
+
+
pecially, variable natural abundances have been observed and isotope enrichment may occur during chemical processing.7 The other carboranes and alkyl carboranes studied were processed in the same fashion.3 They include C Z B ~ H(Table ~ IV), C H ~ C B ~ H(Table B V), (CH3)2C&& (Table VI), (CH3)2C2B7H7 (Table VII), and (CH3)2C2BsH8(Table VIII). The results must be used with the caution that the polyisotopic spectra were published as line drawings. More accurate monoisotopic spectra could be derived from digital data. The Journal of Physical Chemistry, Vol. 76, No. 13,19'79
3 2 s
'2C 14N '60
'H a
170
0.5054 0.7553 0.950 0.9889 0.9963 0,99759 0.99986
81Br 37C1
We must neglect 33S(0.0076) and (0.00037).
3 4 s
'3C '6N '80
2H 3%
0,4946 0,2447 0.04220 0.0111 0,0037 0.00204* 0.00015
(0.00014).
* Omitting
If B,P,M, represents the formulas of compounds which can be handled, then P stands for any one of the polyisotopic elements previously mentioned, while hl stands for any combination of the following (at least approximately) monoisotopic elements: H, F, I, P, Be, Al, As, He, Na, Cs, Co, Au, Bi, Sc, Y, Nb, Rh, La, Tb, Ho, Tm, and Ta. I n addition ions such as C,H,+ in carboranes and HC1+, or Br + in haloboranes, are permitted. Any combination of 2, y, or x may be zero in the formula above. This feature is especially helpful for removing background peaks such as HzO+ or Nz+. Some special note should be taken of the deuterium derivatives of the boranes. The naturally occurring deuterium content of the boranes may be taken into account explicitly and exactly. The effect is usually undetectable, but it becomes more important as the number of hydrogen atoms in the ion increases. I n B20H26, for example, the first isotopic peak has a relative size of 0,00015 X 26 or 0.0039.* Since the largest possible peak is 3 00, the largest contribution possible is 0.39. Hydrogen, consequently, can be treated as a monoisotopic element. Deliberately deuterated boranes can be treated exactly if the deuteration is random and not site-selective. The statistics used to calculate the coefficients of eq 1 assume random occupation of the combinations. Even here the deuterated (8) F. W . McLafferty, "Interpretation of Mass Spectra," W. A . Benjamin, New York, N. Y., 1967, p 210.
THEMONOISOTOPIC MASSSPECTRA OF BORANE DERIVATIVES borane could be accommodated by treating the siteselective deuterium as a monoisotopic element in the ions in which it occurs.
Error Analysis Several limitations on the use of the technique have been noted above or previously published. It was said, for example, that the mass spectrum of B2&6 could not be resolved into a monoisotopic spectrum.6 This was rather hastily attributed to the presence of impurities, though the spectra of other boranes could be resolved despite the presence of nonboron peaks. To study the problem we calculated an arbitrary but exact polyisotopic spectrum for BzoHze and proceeded to find the cause of our previous failure. The neglect of the natural abundance of deuterium was found to be unimportant. The presence of perturbed intensities was no more troublesome than in other molecules. It was noted, however, that the contribution of any one formula to a measured intensity is quite small since each ion containing 20 boron atoms will be spread over 21 separate masses. I n fact, rounding errors are of the same order of magnitude as the contribution of the monoisotopic formula to the measured intensity. With the idea that it would be more sensitive to solve for the most abundant ion rather than for the pure isotope, we wrote a program which solved for intensity of the 17 times more abundant "Bl6"B4 rather than for 'lBzo. This proved to be no improvement. The procedure merely multiplies the coefficients in (1) by a constant, with no effect on the results. Finally, it was discovered that the number of significant figures in the polyisotopic data was the controlling factor. If four significant figures after the decimal are used, good intensities and no negatives are generated. With three places, tolerably accurate intensities are produced. With two places about 1% of the sum of the intensities are negative. One place produces 12% negatives and completely meaningless intensities in the monoisotopic spectrum. Since our previous data had been read from a graph to the nearest integer, the failure to produce an
1867
acceptible monoisotopic spectrum from the experimental information is not s ~ r p r i s i n g . ~The conclusion we must draw is that the monoisotopic mass spectra from boranes with higher molecular weights can only be generated from increasingly more precise polyisotopic intensities. Finally one might mention some limitations. First of all, not all pdyisotopic mass spectra can be resolved into monoisotopic intensities by any method. Whenever there are more monoisotopic peaks to be established than there are polyisotopic measurements to determine them, the problem is mathematically insoluble. For example if BD3+, CD2+, and K D + are present simultaneously, all isotopic variants will occur a t masses 16 and 17. Measurements a t these two masses are insufficient to determine the three monoisotopic intensities. For the sake of argument we have assumed that the mass spectrum is a low resolution measurement, and no 'H is present. Secondly, some polyisotopic mass spectra cannot be resolved into monoisotopic spectra even though in theory they fulfill the mathematical criterion mentioned above. If a sufficient number of polyisotopic peaks do not extend above the detection threshold of the mass spectrometer in question, then the experimenter as a matter of fact is unable to resolve the spectrum with the data available. A third limitation occurs when we attempt to resolve the mass spectrum of two similar complex ions. Each ion determines a column of coefficients in the set of simultaneous equations described by (1). If these columns are quite similar, the two columns of the matrix of coefficients are not mathematically independent, and the least-squares matrix method cannot be carried out. This is particularly true if both monoisotopic formulas occur a t the same mass. If BsHll+ and Be+, both at mass 66, are part of a spectrum, their similar coefficients prevent simultaneous solution; one or the other must be chosen. For simpler ions this limitation is not important. L. H. Hall and W. S. Koski, J . Amer. Chem. Soc., 84, 4205 (1962).
(9)
The Journal
of
Physical Chemistry, Vol. 76, No. IS, 1978
