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The LG-CP buildup curves of the carbon signals in three- and four spin systems

Here we calculate the LG-CP buildup curves of the carbon signals in a three-spin system and in two four-spin systems and . In the derivation we will neglect all off-resonance terms in the average Hamiltonian of these systems. Their influence is discussed in the main text.

The three-spin system

The average interaction Hamiltonian of this system has the form

(A1)

and can be conveniently represented in the following basis set of product functions

, (A2)

resulting in the matrix

. (A3)

In terms of the standard fictitious spin-1/2 operators this has the form

. (A4)

Assuming real and positive and coefficinents, a straightforward diagonalization of this Hamiltonianyields a set of eigenvalues

(A5)

with

(A6)

and an eigenvector matrix

, (A7)

where we introduce the following notations

. (A8)

Hence, for the -angle is 0 and , and for the -angle is and . In the new basis set, represented by the columns of matrix D, the polarization operators , and become

(A9)

and

. (A10)

The proton and the carbon and signals are proportional to the expectation values of these operators. Because the diagonalized Hamiltonian has the form

, (A11)

only the terms in Eq. (A9) contribute to the time-dependent parts of the signals. Starting with the initial state represented by the initial proton polarization, the signals can easily be calculated according to , where O represents either S1z or S2z of Eq. (A10)

. (A12)

Here the represent the intensities of the carbon signals after a single excitation pulse. Some properties of these solutions are discussed in the main text.

The four-spin system

The LG-CP buildup curves of the carbon signals in will change when an additional proton is added to the system. An exact analytical solution is not possible in this case and perturbation theory must be used to obtain expressions for the polarization transfer functions for both carbons.

We will first consider two coupled groups. The effective dipolar Hamiltonian of this spin system has four heteronuclear dipolar terms, namely

(A13)

with i=1,2 for the protons and j=1,2 for the carbons, and has to be diagonalized in order to evaluate the LG-CP carbon signals. We represent the Hamiltonian in a basis set of product states

Assuming that and , we first diagonalize the and terms of the Hamiltonian. This diagonalization leaves the and matrix elements off-diagonal.

After the diagonalization of the part of the Hamiltonian in Eq. (A13), the spin states mix and the basis set becomes

(A14)

with and .

The diagonal terms of can be written as

(A15)

with diagonal elements

. (A16)

In this set of functions the proton and carbon polarizations become

(A17)

and the and terms of the Hamiltonian remain off-diagonal

(A18)

with and . As expected, if all off-diagonal terms of are ignored, the diagonal elements in Eq. (A15) will generate an oscillatory time dependence of the operators in the expressions for , with the frequency 2d11 or 2d22. These terms will result in a nonzero signal. This case will simply correspond to two groups, which evolve independently:

. (A19)

If the Hamiltonian can be approximated in the following way. To first order only terms connecting nearly degenerate states, defined in Eq. (A15), must be retained. The influence of the neglected terms will be on the order of d12/d11. The simplified off-diagonal Hamiltonian will then look like

. (A20)

The approximated Hamiltonian can be factorized into 7 submatrices: {a}, {p}, {b,c,e,i},{f},{k},{d,g,j,m}, and {h,l,n,o}. To simplify the discussion, we will assume that, and(). Then the submatrices can be easily diagonalized. The diagonal elements in Eq. (A16) are drawn schematically in FigureA1 together with the matrix elements of in Eq. (A20) and the coherences of the and polarizations in Eq. (A17).The diagonalization of the full Hamiltonian results in some additional splitting of the energy levels. Moreover, it produces some changes in the expressions for the I1z, S1z, and S2zpolarizations. A simple calculation then results in the following carbon signals, originating from the proton

. (A21)

The four-spin system

The matrix representation of the Hamiltonian of the spin system, consisting of a group coupled to a second carbon, has again a dimension of 16x16. In order to choose a convenient spin basis set to describe the LG-CP process, we will first discuss the LG-CP buildup curve of ignoring 13C(2) (d12=d22=0). For simplicity we will assume here that the proton-carbon interactions are equal. It is convenient to choose triplet and singlet proton states

(A22)

and to construct the total basis set

. (A23)

Here the subscripts represent the projection M of the total spin of each proton state. In this representation the Hamiltonian has only matrix elements between states with equal M-values ()

. (A24)

Assuming a group in which the protons have the same chemical shift and are spectroscopically indistinguishable, the total proton (both protons are initially polarized) and the carbon polarization operators in the basis set of Eq. (A23) can be represented as

. (A25)

Clearly, only the first term of the initial state can become time-dependent due to in Eq. (A24). The triple-quantum and the mixed M-state operators do not result in any observable and the signals of the protons and carbon become

. (A26)

The normalized proton signal oscillates between the values 1 and 1/2, whereas the carbon reaches a maximum value of . This result is only valid when the dipolar interactions between the protons and the carbon are equal. In real systems these values are different and the Hamiltonian in Eq. (A24) exhibits an additional term with an operator , which mixes triplet and singlet proton states. This makes the calculation more cumbersome, and we must rely on numerical simulations.

The polarization dynamics in a spin system of the form is of course more complicated and here we will restrict ourselves to the special case in which the couplings between the protons and are again equal and much stronger than the (equal) interactions between the protons and . The general form of the matrix representation of the LG-CP Hamiltonian (with real dipolar coefficients) of this four-spin system can be represented in a basis set composed of the 16 states

. (A27)

In this basis set the three-spin states are defined in Eq. (A23) for and and are the spin states of . According to these and states the Hamiltonian can be written in terms of four 8x8 matrices

. (A28)

The non-zero matrix elements of the two block diagonal parts of depend on and become

(A29)

The off-diagonal elements, proportional to , are

. (A30)

The three lines of matrix elements in Eqs. (A29) and (A30) connect states inside the manifolds, respectively. The matrix representations of the M-manifolds of show that when , the -elements in the first (M=1) and the third (M=-1) lines of Eq. (A29) can be ignored, whereas the elements in the second (M=0) line must be retained. The approximate Hamiltonian then takes the following form

. (A31)

This Hamiltonian consists of two commuting terms that operate on different parts of the initial proton population

. (A32)

The first two terms become time-dependent due to in and the third term due to . Because the z-components of the S-spin angular momentum operators can be written as

, (A33)

the LG-CP signals of the two carbons become

. (A34)

The strongly coupled carbon signal oscillates between 0 and and the weakly coupled carbon between 0 and . Powder integration will generate partial polarizations of two carbons with a 2:1 ratio between the average values of the signals at long times . In real spin systems, where the symmetry condition imposed on the Hamiltonian by the equal dipolar interaction strengths is disturbed, the spin dynamics are more complicated than in Eq. (A34) and mixing between the triplet and singlet proton states changes the relative leveling-off values of the buildup curves.

EPAPS figure

Figure A1.

Energy level diagram of the four-spin {13C(1) H(1)- 13C(2)H(2)} system. The product spin states (a-p) are defined in Eq. (A14). The solid and the dashed arrows show the coherences of the and polarizations in Eq. (A17), respectively. The direction of the arrows corresponds to the sign of the represented operator for real positive dipolar coefficients. The thick filled two-directional arrows show the coherences with a coefficients in the Hamiltonian in Eq. (A20) and thick opaque two-directional arrows show the -coherences. These elements will result in additional splitting of the energy levels and will produce changes in the expressions for spin polarizations.