Direct Determination of Pure-State Density Matrices. V. Constrained Eigenvalue Problems

Fuzzy Electron‐Density Fragments as Building Blocks in Crystal‐Engineering Design

“…The fact that quantum-chemical representations of molecular electron densities may provide substantial help in the actual evaluation and analysis of X-ray diffraction experiments has received a new confirmation by the introduction of quantum crystallography [1,2], itself a combination of crystallographic and quantum-chemical approaches. The developments leading to quantum crystallography include the early approaches of fitting N-representable density matrices, alternatively, fitting actual molecular wavefunctions to experimentally determined X-ray diffraction data [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. One of the goals of quantum crystallography is finding efficient ways to utilise crystallographic diffraction data for the determination of molecular density matrices or molecular wavefunctions.…”

The cusp condition: Constraint on the electron density matrix

“…Hence, the further usefulness of this equation to describe larger molecules depends upon the possibility of finding more conditions whose imposition on the density results in an improvement of molecular properties. In addition to those molecular properties [3] already studied in detail in connection with Equation (2), three types of nonempirical conditions present themselves ; hypervirials [4], local energy [ 5 ] , and cusp conditions [6]. Of these, only the first two provide sufficient constraints to determine any density matrix.…”

“…A recent series of papers [3] has detailed the derivation of this equation, studied its properties, and exploited it in the calculation of properties of a number of diatomic molecules. Of course, in the use of Equation (2), the number of constraints required to fix the elements of the density matrix uniquely increases with the number of base functions.…”

A constrained Roothaan–Hartree–Fock method

“…Each Hartree-Fock one-electron function 4, is expanded in terms of a finite number of known one-electron functions xi where m > n, and the coefficients ci, are to be determined. Substituting 4, into the free variational principle gives the constrained Roothaan-Hartree-Fock equations, which in canonical form are represented by the pseudoeigenvalue equation Equations analagous to [3]- [5] for open shell systems in either the unrestricted (34) or restricted (35) formulations are easily formulated.…”