A simple natural orbital mechanism of “pure” van der Waals interaction in the lowest excited triplet state of the hydrogen molecule

A treatment of van der Waals (vdW) interaction by density-matrix functional theory requires a description of this interaction in terms of natural orbitals (NOs) and their occupation numbers. From an analysis of the configuration-interaction (CI) wave function of the <SUP>3</SUP>Sigma<SUB>u</SUB><SUP>+</SUP> state of H<SUB>2</SUB> and the exact NO expansion of the two-electron triplet wave function, we demonstrate that the construction of such a functional is straightforward in this case. A quantitative description of the vdW interaction is already obtained with, in addition to the standard part arising from the Hartree-Fock determinant |1sigma<SUB>g</SUB>(r<SUB>1</SUB>)1sigma<SUB>u</SUB>(r<SUB>2</SUB>)|, only two additional terms in the two-electron density, one from the first “excited” determinant |2sigma<SUB>g</SUB>(r<SUB>1</SUB>)2sigma<SUB>u</SUB>(r<SUB>2</SUB>)| and one from the state of <SUP>3</SUP>Sigma<SUB>u</SUB><SUP>+</SUP> symmetry belonging to the (1pi<SUB>g</SUB>)<SUP>1</SUP>(1pi<SUB>u</SUB>)<SUP>1</SUP> configuration. The potential-energy curve of the <SUP>3</SUP>Sigma<SUB>u</SUB><SUP>+</SUP> state calculated around the vdW minimum with the exact density-matrix functional employing only these eight NOs and NO occupations is in excellent agreement with the full CI one and reproduces well the benchmark potential curve of Kolos and Wolniewicz [J. Chem. Phys. 43, 2429 (1965)]. The corresponding terms in the two-electron density rho<SUB>2</SUB>(r<SUB>1</SUB>,r<SUB>2</SUB>), containing specific products of NOs combined with prefactors that depend on the occupation numbers, can be shown to produce exchange-correlation holes that correspond precisely to the well-known intuitive picture of the dispersion interaction as an instantaneous dipole-induced dipole (higher multipole) effect. Indeed, (induced) higher multipoles account for almost 50% of the total vdW bond energy. These results serve as a basis for both a density-matrix functional theory of van der Waals bonding and for the construction of orbital-dependent functionals in density-functional theory that could be used for this type of bonding.