Evaluation of matrix exponentials

The expansion of the exponential matrix (3.1.2) is rapidly convergent and may often be used for the evaluation of unitary matrices, especially if the anti-Hermitian matrix X has a small norm and high accuracy is not required. An alternative strategy is to diagonalize X  

The exponential of the pure imaginary diagonal matrix i8 with complex elements i8jt is easily calculated. The final matrix U is then obtained after a few matrix multiplications. 

The same method may be used for special orthogonal matrices (3.1.25). Diagonalization of the real antisymmetric matrix X gives 

Note thaL even though R and are both real, the evaluation of (3.1.29) involves complex arithmetic since V is complex and it imaginary. 

To see how the real orthogonal matrix R may be obtained using only real arithmetic, we note that the square of X may be diagonalized by an orthogonal matrix 

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In DM-MP2 calculations, we applied the Chebyshev expansion for the evaluation of matrix exponential, which is implanented in the Expoktt library program [51]. For the numerical quadrature of the Laplace-transformed integrals of Eqs. (9) and (27), we used the r-point Euler-Maclaurin (trapezoidal) quadrature... 

The transformation (10.7.32) appears to be a rather complicated one, requiring the evaluation of matrix exponentials. However, as discussed in Section 3.1.6, we may write the transformed density matrix as an asymmetric BCH expansion... 

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