This paper is part of the 2012 Quantum Chemistry thematic issue.

Very accurate variational nonrelativistic calculations are performed for the five lowest Rydberg 2 D states (1s 2 nd 1 , n = 3, . . . ,7) of the lithium atom ( 7 Li). The finite-nuclear-mass approach is employed and the wave functions of the states are expanded in terms of all-electron explicitly correlated Gaussian function. Four thousand Gaussians are used for each state. The calculated relative energies of the states determined with respect to the 2 S 1s 2 2s 1 ground state are systematically lower than the experimental values by about 2.5 cm −1 . As this value is about the same as the difference between the experimental relative energy between 7 Li + and 7 Li in their ground-state energy and the corresponding calculated nonrelativistic relative energy, we attribute it to the relativistic effects not included in the present calculations.

Very accurate variational non-relativistic calculations are performed for four higher Rydberg 2 D states (1s 2 nd 1 , n = 8, . . . , 11) of the lithium atom ( 7 Li). The wave functions of the states are expanded in terms of all-electron explicitly correlated Gaussian functions and finite nuclear mass is used. The exponential parameters of the Gaussians are optimized using the variational method with the aid of the analytical energy gradient determined with respect to those parameters. The results of the calculations allow for refining the experimental energy levels determined with respect to the 2 S 1s 2 2s 1 ground state.

Variational calculations of ground and excited bound states on atomic and molecular systems performed with basis functions that explicitly depend on the interparticle distances can generate very accurate results provided that the basis function parameters are thoroughly optimized by the minimization of the energy. In this work we have derived the algorithm for the gradient of the energy determined with respect to the nonlinear exponential parameters of explicitly correlated Gaussian functions used in calculating n-electron atomic systems with two p-electrons and ͑n −2͒ s-electrons. The atomic Hamiltonian we used was obtained by rigorously separating out the kinetic energy of the center of mass motion from the laboratory-frame Hamiltonian and explicitly depends on the finite mass of the nucleus. The advantage of having the gradient available in the variational minimization of the energy is demonstrated in the calculations of the ground and the first excited 3 P state of the carbon atom. For the former the lowest energy upper bound ever obtained is reported.

An algorithm for the variational calculation of atomic D states employing n-electron explicitly correlated Gaussians is developed and implemented. The algorithm includes formulas for the first derivatives of the Hamiltonian and overlap matrix elements determined with respect to the Gaussian nonlinear exponential parameters. The derivatives are used to form the energy gradient which is employed in the variational energy minimization. The algorithm is tested in the calculations of the two lowest D states of the lithium and beryllium atoms. For the lowest D state of Li the present result is lower than the best previously reported result.