Analytic Energy Gradients and Hessians of Exact Two-Component Relativistic Methods: Efficient Implementation and Extensive Applications

Abstract

The algebraic exact two-component (X2C) relativistic Hamiltonian can be viewed as a matrix functional of the decoupling (X) and renormalization (R) matrices. It is precisely their responses to external perturbations that render X2C-based response theories different in form from the nonrelativistic counterparts. However, the situation is not really bad. Sticking to the energy gradients, it can be shown that the nuclear derivatives of X and R (X μ and R μ , respectively) can be transformed away to favor transformed, nucleus-independent density matrices, viz., the X2C energy gradients can be written in a form that does not depend explicitly on X μ and R μ . Further combined with the storage of quantities that are already available in the energy calculation, only 35 matrix multiplications are needed to construct the one-electron (relativistic) part of the X2C gradients, thereby rendering the gradient calculations very efficient. More efficiency can be gained by approximating the molecular X as the superposition of the atomic ones (denoted as X2C/AXR) and by further approximating the molecular R also as the superposition of the atomic ones (denoted as X2C/AU): The numbers of matrix multiplications required for constructing the one-electron (relativistic) parts of the X2C/AXR and X2C/AU gradients are reduced to 18 and 4, respectively. Similar approximations can also be applied to the X2C Hessian. It will be shown numerically that the X2C/AXR gradients and Hessians are extremely accurate (almost indistinguishable from the full X2C ones), whereas the X2C/AU ones do have discernible errors but which are tolerable in view of the dramatic gain in efficiency.

NMR Indirect Spin–Spin Coupling Constants in a Modern Quasi-Relativistic Density Functional Framework

“…The derivatives of the decoupling matrix, X u M , involve the perturbed coefficients, C u S, M and C u L, M , and are therefore evaluated using one-electron response theory. − , Using X u M , the derivative of the renormalization matrix is evaluated by solving Sylvester matrix equations. , The evaluation of the derivatives of X and R is described in Section . The second derivative reads with the derivative of the NESC matrix Again, the derivatives of X and R are obtained by solving (one-electron) response equations and Sylvester matrix equations, respectively. − , We note in passing that the derivatives can be formulated without explicitly calculating the derivative of X . In both cases, local approximations are useful to reduce the computational demands. ,,, …”

“…[52][53][54]69 We note in passing that the derivatives can be formulated without explicitly calculating the derivative of X. 57 In both cases, local approximations are useful to reduce the computational demands. 56,57,69,90 From a formal point of view, the RMB condition 19,107,108 should be used as it ensures the exact nonrelativistic limit of the Dirac equation in the presence of a vector potential.…”

“…57 In both cases, local approximations are useful to reduce the computational demands. 56,57,69,90 From a formal point of view, the RMB condition 19,107,108 should be used as it ensures the exact nonrelativistic limit of the Dirac equation in the presence of a vector potential. The RMB condition ensures the variational stability to − c ( ) 4 in the presence of magnetic fields.…”

“…It was later shown that the exact decoupling of the positronic and electronic states can be achieved in one shot with the so-called exact two-component (X2C) theory, − which was pioneered by Dyall in the context of the normalized elimination of the small component (NESC). − The decoupling is achieved by a single diagonalization of the Dirac Hamilton operator in a matrix representation. This allows for a rather straightforward computation of analytical derivatives for geometric, − electric, − and magnetic − first- or second-order molecular properties. Nowadays, X2C is implemented in many quantum chemistry program suites − and is an established Hamiltonian for molecular properties.…”

Relativistic quantum chemistry: today and tomorrow

“…然而, R + 的局域性比X差得多, 这种近似必 然引入一定的误差 [91] . 虽然如此, 相对于计算效率的 大幅提高(尤其是对解析梯度和二阶导数 [86] ), 这些误 差是勉强可以接受的. 很显然, 当重原子间距很近时 (<1.5 Å), 对X (以及R + )的原子近似可以推广到双原子 (分子片)近似 [38,41] .…”