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Lineshape function for absorption-mode DNMR spectrum
JOURNAL
OF MAGNETIC
RESONANCE
16,182-184 (1974)
LineshapeFunction for Absorption-Mode DNMR Spectrum s. SZYMArjSKI Institute of Organic Chemistry, Polish Academy of Sciences, 01-224 Warszawa, Poland
AND A. GRYFF-KELLER Institute of Organic Chemistry and Technology, Polytechnical University, 00-661 Warszawa, Polarrd
ReceivedMay 14,1974 A new way of calculating NMR absorption for intramolecularly exchanging systemsis proposed. Symmetrization of the exchangematrix and extraction of the absorption part from the complex lincshapefunction implies that the calculation of the absorption spectrumconsistsin a decomposition,at everypoint o, of the real, symmetric, and positive definite matrix into a product of triangular matrices. It seemsthat the computer program ASESbased on this algorithm, written in Algol, may in some casessuccessfullycompete with programs in which the method of diagonalization of the complex matrix is used. The complex DNMR lineshape function for the unsaturated steady-state spectrum of an intramolecularly exchanging spin system may be expressed as a sum of terms belonging to different sets of eigenvalues of the superoperators which are invariant under chemical exchange (1-3) :
M(o) = -iC;{F;*[i(L,
- wl) + R, + Xl-‘P, c;}~.
PI
For simplicity, only one term will be discussed. In Eq. [l], o = 21rv is the frequency of the HI field, C is a positive real number (scaling factor), i2 = -1,1 is the unit matrix, the asterisk signifies Hermitian conjugation; matrices: L, are Zeeman and spin-spin interactions, R, is the relaxation, P, are populations; vectors: F; is the lowering operator and ur (=F; under considered conditions). The quantities with a “c” subscript are the direct sums of matrices or vectors describing particular spin configurations. The X matrix describes chemical exchange. If, as usual, the basis of spin product functions is adopted, L, and R, matrices are real and symmetrical and the F; vector is the direct sum of identical F- vectors with elements 0 for combination transitions or 1 otherwise. The X matrix consists of rz square blocks (r is the number of spin configurations) :
where Kp4 is the constant of first-order rate interchange from cotiguration “9” (I). Copyright0
1974 by Academic Press, Inc. AU rights of reproduction in any form reserved. Printed in Great Britain
182
“~3’ to
LINESHAPE
FOR
DNMR
183
ABSORPTION
Disadvantages due to the lack of symmetry of the matrix may be removed easily by transforming Eq. [l] with matrix Pil2 (PC is the direct sum of r blocks ~~1, where PI is the population of the ith spin configuration): M(w) = -iCF;* P;‘z PT~‘~[i(L, - 01) + R, f Xl-’ P;” P;” a; := -iCS* [i(Lc - Of) - Y]-’ S,
VI
where S = P:/2F; = Pi/2n& because the P, matrix commutes with all the blockdiagonal matrices. Matrix Y = -R, + P;l’* (-X)Pi” is symmetrical due to the equilibrium conditions : pi Ki j = p, K,,. Matrix P;1i2(-X)P:/2 consists of blocks : [P,“2(-mX)P~‘2]i~ = -Xii,
[P~1’2(-X)PE’2]~j =
-(Kji
Kij)l” 1.
The symmetry of this matrix reflects the fact that for a description of the dynamic equilibrium between r states, r(r + 1)/2 parameters are needed. Due to the symmetrization of the matrix, a method for the computation of a DNMR absorption spectrum, different from the diagonalization procedure (I, 4, 5), may be proposed. Applying the matrix identity : (A + Bi)-’ = (BA-’ B + A)-l - iA-’ B(BA-’ B + A)-‘, 131 the absorption function M,(W) (the imaginary part) may be extracted from Eq. [2] : M,(o) = cs*[(L,
- wl)Y-‘&
- wl) + Y]-?3.
[41 The identity may be used provided that the inverse of the Y matrix and the reciprocal of the matrix in square brackets exist. The relaxation matrix R, in the completely general case is negative semidefinite (6), but in. practice it is assumed to be nonsingular and often diagonal (7). Simultaneously, all the eigenvalues of the X matrix are nonpositive (Levy-Hadamard theorem, Ref. 8), i.e., the P-1’2(-X)P1’2 matrix is positive semidefinite. Thus, the Y matrix may be considered as positive definite. The latter implies that the matrix in square brackets in Eq. [4] may be rewritten as Y1’2{[Y-1’2(L, - d)Y-q* + 1}Y”2. From, this, it is evident that the matrix is positive definite too. Thus, indeed, identity [3] is applicable. The form of Eq. [4] implies the absorption is always positive, which is not surprising. For computational purposes, it seems convenient to transform Eq. [4] with unitary matrix U in such a manner that the only nonzero element of the S’ = US vector is the last one. The last row of the U matrix ought to be equal to (S*S)-lj2S*. Now the absorption is expressed by M,(o) = C,[(A - oB + d Y-1)-1],,,
PI
where A = LiY’-“L: + Y’, B = Y’-lLi + LLY’-‘, C1 = C(S*S), T’ = UTU* for all matrices, and n is the matrix size. It seems that the simplest method for a computation of absorption is to express the real, symmetric, and positive-definite matrix in round brackets in Eq. [5] as a product of triangular matrices NNT (8). Then KG4 = G(NJ2, where N is the lower triangular matrix.
PI
184
SZYMAtiSKI
AND
GRYFF-KELLER
If, as usual, a trivial relaxation mechanism is assumed where R, = -(l/T,)l, then the U matrix may be expressed as the product of two unitary transformations U = QV. The V matrix diagonalizes Y and may have a Y-like block structure, where the blocks vr, = PilJ2 1 for i = 1, 2, . . ., r. All the eigenvectors forming this block are connected with the eigenvalue l/T,. For example, for a two-site exchange, V,, = V,, =pi121, VI2 =-Vzl = -p:“l, Y;, = (l/T2 + Ki2 + K,,)l, Yi2 = (l/T,)l. The Q matrix may be a direct sum of blocks-unitary matrices :
Q = diag(ll, 12, . . ., L 1,QA where the last row of Qr is equal to (F-*F-)-1/2F-*. The above procedure, performed once per spectrum, reduces the computational effort in every point o. To check the efficiency of the algorithm just described, we prepared computer program ASES, written in Algol, to simulate DNMR absorptions. The factorization due to the invariance under chemical exchange of the “z” component of the total spin is included. Additionally, the factorization connected with the invariance of the “z” component of the X part of the spin system as well as one resulting from the one-element symmetry may be exploited when it is allowed by the nature of the exchanging system. For a few tested cases, the results obtained are identical with those from Bins&s DNMR (I) program. For a computation of the lineshape for an ABC + DEF system, only about 6K of computer space is needed. It is difficult to compare the speeds of computation because we work with a second-generation computer ODRA 1204. However, it seems that more than 400 spectral points may be computed with ASES in the same time that would be necessary for a diagonalization of the complex matrix in Binsch’s program in the same computer system. It appears that this computation of DNMR absorption may compete with the method of the diagonalization of the complex matrix especially when only some parts of the spectrum or only some points are needed. ACKNOWLEDGMENT The authors are greatly indebted to Dr. M. Witanowski for help in preparing this paper. REFERENCES G. BINSCH,J. Amer. Chem. Sot. 91,1304 (1969). D. A. KLEIER AND G. BINSCH, J. Magn. Resonance 3,146 (1970). A. GRYFF-KELLER AND S. SZYMA~KI, Rocz. Chem. 47,1889 (1973). R. G. GORDON AND R. P. MCGINNIS, J. Chem. Phys. 49,2455 (1968). L. W. REEVESAND K. N. SHAW, Can. J. Chem. 48,3641(1970). R. A. HOFFMAN, “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Vol. 4, p. 88, Academic Press, New York, 1970. 7. G. BINSCH, Mol. Phys. 15,469 (1968). 8. E. B~EDWIG, “Matrix Calculus” (2nd ed.), pp. 67, 49, North-Holland Pub]. Comp., Amsterdam, 1959.
1. 2. 3. 4. 5. 6.
OF MAGNETIC
RESONANCE
16,182-184 (1974)
LineshapeFunction for Absorption-Mode DNMR Spectrum s. SZYMArjSKI Institute of Organic Chemistry, Polish Academy of Sciences, 01-224 Warszawa, Poland
AND A. GRYFF-KELLER Institute of Organic Chemistry and Technology, Polytechnical University, 00-661 Warszawa, Polarrd
ReceivedMay 14,1974 A new way of calculating NMR absorption for intramolecularly exchanging systemsis proposed. Symmetrization of the exchangematrix and extraction of the absorption part from the complex lincshapefunction implies that the calculation of the absorption spectrumconsistsin a decomposition,at everypoint o, of the real, symmetric, and positive definite matrix into a product of triangular matrices. It seemsthat the computer program ASESbased on this algorithm, written in Algol, may in some casessuccessfullycompete with programs in which the method of diagonalization of the complex matrix is used. The complex DNMR lineshape function for the unsaturated steady-state spectrum of an intramolecularly exchanging spin system may be expressed as a sum of terms belonging to different sets of eigenvalues of the superoperators which are invariant under chemical exchange (1-3) :
M(o) = -iC;{F;*[i(L,
- wl) + R, + Xl-‘P, c;}~.
PI
For simplicity, only one term will be discussed. In Eq. [l], o = 21rv is the frequency of the HI field, C is a positive real number (scaling factor), i2 = -1,1 is the unit matrix, the asterisk signifies Hermitian conjugation; matrices: L, are Zeeman and spin-spin interactions, R, is the relaxation, P, are populations; vectors: F; is the lowering operator and ur (=F; under considered conditions). The quantities with a “c” subscript are the direct sums of matrices or vectors describing particular spin configurations. The X matrix describes chemical exchange. If, as usual, the basis of spin product functions is adopted, L, and R, matrices are real and symmetrical and the F; vector is the direct sum of identical F- vectors with elements 0 for combination transitions or 1 otherwise. The X matrix consists of rz square blocks (r is the number of spin configurations) :
where Kp4 is the constant of first-order rate interchange from cotiguration “9” (I). Copyright0
1974 by Academic Press, Inc. AU rights of reproduction in any form reserved. Printed in Great Britain
182
“~3’ to
LINESHAPE
FOR
DNMR
183
ABSORPTION
Disadvantages due to the lack of symmetry of the matrix may be removed easily by transforming Eq. [l] with matrix Pil2 (PC is the direct sum of r blocks ~~1, where PI is the population of the ith spin configuration): M(w) = -iCF;* P;‘z PT~‘~[i(L, - 01) + R, f Xl-’ P;” P;” a; := -iCS* [i(Lc - Of) - Y]-’ S,
VI
where S = P:/2F; = Pi/2n& because the P, matrix commutes with all the blockdiagonal matrices. Matrix Y = -R, + P;l’* (-X)Pi” is symmetrical due to the equilibrium conditions : pi Ki j = p, K,,. Matrix P;1i2(-X)P:/2 consists of blocks : [P,“2(-mX)P~‘2]i~ = -Xii,
[P~1’2(-X)PE’2]~j =
-(Kji
Kij)l” 1.
The symmetry of this matrix reflects the fact that for a description of the dynamic equilibrium between r states, r(r + 1)/2 parameters are needed. Due to the symmetrization of the matrix, a method for the computation of a DNMR absorption spectrum, different from the diagonalization procedure (I, 4, 5), may be proposed. Applying the matrix identity : (A + Bi)-’ = (BA-’ B + A)-l - iA-’ B(BA-’ B + A)-‘, 131 the absorption function M,(W) (the imaginary part) may be extracted from Eq. [2] : M,(o) = cs*[(L,
- wl)Y-‘&
- wl) + Y]-?3.
[41 The identity may be used provided that the inverse of the Y matrix and the reciprocal of the matrix in square brackets exist. The relaxation matrix R, in the completely general case is negative semidefinite (6), but in. practice it is assumed to be nonsingular and often diagonal (7). Simultaneously, all the eigenvalues of the X matrix are nonpositive (Levy-Hadamard theorem, Ref. 8), i.e., the P-1’2(-X)P1’2 matrix is positive semidefinite. Thus, the Y matrix may be considered as positive definite. The latter implies that the matrix in square brackets in Eq. [4] may be rewritten as Y1’2{[Y-1’2(L, - d)Y-q* + 1}Y”2. From, this, it is evident that the matrix is positive definite too. Thus, indeed, identity [3] is applicable. The form of Eq. [4] implies the absorption is always positive, which is not surprising. For computational purposes, it seems convenient to transform Eq. [4] with unitary matrix U in such a manner that the only nonzero element of the S’ = US vector is the last one. The last row of the U matrix ought to be equal to (S*S)-lj2S*. Now the absorption is expressed by M,(o) = C,[(A - oB + d Y-1)-1],,,
PI
where A = LiY’-“L: + Y’, B = Y’-lLi + LLY’-‘, C1 = C(S*S), T’ = UTU* for all matrices, and n is the matrix size. It seems that the simplest method for a computation of absorption is to express the real, symmetric, and positive-definite matrix in round brackets in Eq. [5] as a product of triangular matrices NNT (8). Then KG4 = G(NJ2, where N is the lower triangular matrix.
PI
184
SZYMAtiSKI
AND
GRYFF-KELLER
If, as usual, a trivial relaxation mechanism is assumed where R, = -(l/T,)l, then the U matrix may be expressed as the product of two unitary transformations U = QV. The V matrix diagonalizes Y and may have a Y-like block structure, where the blocks vr, = PilJ2 1 for i = 1, 2, . . ., r. All the eigenvectors forming this block are connected with the eigenvalue l/T,. For example, for a two-site exchange, V,, = V,, =pi121, VI2 =-Vzl = -p:“l, Y;, = (l/T2 + Ki2 + K,,)l, Yi2 = (l/T,)l. The Q matrix may be a direct sum of blocks-unitary matrices :
Q = diag(ll, 12, . . ., L 1,QA where the last row of Qr is equal to (F-*F-)-1/2F-*. The above procedure, performed once per spectrum, reduces the computational effort in every point o. To check the efficiency of the algorithm just described, we prepared computer program ASES, written in Algol, to simulate DNMR absorptions. The factorization due to the invariance under chemical exchange of the “z” component of the total spin is included. Additionally, the factorization connected with the invariance of the “z” component of the X part of the spin system as well as one resulting from the one-element symmetry may be exploited when it is allowed by the nature of the exchanging system. For a few tested cases, the results obtained are identical with those from Bins&s DNMR (I) program. For a computation of the lineshape for an ABC + DEF system, only about 6K of computer space is needed. It is difficult to compare the speeds of computation because we work with a second-generation computer ODRA 1204. However, it seems that more than 400 spectral points may be computed with ASES in the same time that would be necessary for a diagonalization of the complex matrix in Binsch’s program in the same computer system. It appears that this computation of DNMR absorption may compete with the method of the diagonalization of the complex matrix especially when only some parts of the spectrum or only some points are needed. ACKNOWLEDGMENT The authors are greatly indebted to Dr. M. Witanowski for help in preparing this paper. REFERENCES G. BINSCH,J. Amer. Chem. Sot. 91,1304 (1969). D. A. KLEIER AND G. BINSCH, J. Magn. Resonance 3,146 (1970). A. GRYFF-KELLER AND S. SZYMA~KI, Rocz. Chem. 47,1889 (1973). R. G. GORDON AND R. P. MCGINNIS, J. Chem. Phys. 49,2455 (1968). L. W. REEVESAND K. N. SHAW, Can. J. Chem. 48,3641(1970). R. A. HOFFMAN, “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Vol. 4, p. 88, Academic Press, New York, 1970. 7. G. BINSCH, Mol. Phys. 15,469 (1968). 8. E. B~EDWIG, “Matrix Calculus” (2nd ed.), pp. 67, 49, North-Holland Pub]. Comp., Amsterdam, 1959.
1. 2. 3. 4. 5. 6.