Natural Densitals

The concept of natural densitals (NDs) and their amplitudes is introduced. These quantities provide the spectral decomposition of the cumulant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math> of the two-electron density that, by definition, quantifies the extent of electron correlation. Consequently, they are ideally suited for a rigorous description of electron correlation effects in Coulombic systems. Spin-summed and spin-resolved versions of the NDs and their amplitudes are defined, and their properties are discussed in detail. Unlike the nonnegative-valued occupation numbers of the natural orbitals (NOs), these amplitudes exhibit diverse sign patterns that emerge within different regimes of electron correlation. The descriptive power of this property is vividly illustrated with the ground state of the H₂ molecule, in which the subtle interplay of various types of electron correlation is captured in detail by a straightforward examination of the amplitudes of the NDs alone. Offering the most compact bilinear representations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math> (a property analogous to that of the NOs with respect to the 1-matrix), the NDs open up entirely new vistas in the analysis of electronic structures of atoms and molecules.