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Spectral Methods in Chemistry and Physics. Bernard Shizgal. Advances in Chemical Physics. Design of Self-Assembling Materials. Ivan Coluzza. Supramolecular Chemistry. The HF energy is given by integrals of molecular orbitals over the Te and Vne operators, often collected in a one-electron operator h, and over the two-electron Vee operator. The latter gives rise to two contributions: a Coulomb term corresponding to the classical interaction between two charge clouds and an exchange term arising from the wave function antisymmetry.
Setting the first derivative of the energy with respect to the molecular orbitals to zero gives the HF equation, which has the same form as the Schrodinger equation, except that it is now at the orbital one-electron level.
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Ff 1f The Fock operator F describes the motion of one electron in the field of all the nuclei and the mean-field of all the other electrons. Since the latter is given by the An Introduction to the State of the Art in Quantum Chemistry 5 orbitals, the HF equation depends on its own solutions and must be solved iteratively. In practical applications, the molecular orbitals are expanded in a basis set, thereby transforming the HF equations into the RoothaanHall equations, which can be written as a generalized matrix eigenvalue problem.
FC 1SC The variational parameters are the molecular orbital coefficients contained in the C matrix which, together with the basis functions, determine the shape of the molecular orbitals. The dimension of the matrix equation is the number of the basis functions, a quantity that is under user control.
The elements in the F matrix contain integrals of the one- and two-electron operators over basis functions, multiplied with products of the molecular orbital coefficients collected into a density matrixD. The iterative sequence corresponds to diagonalization of the Fockmatrix to give an updated density matrix, which is used for constructing the next Fock matrix, etc. The iteration is started with a suitable guess of the density matrix and continued until the difference between two consecutive density matrices are within a suitable small threshold.
At this point, the solution corresponds to a self- consistent field SCF , i. There are two major computational problems in a HF calculation, calculating the two-electron integrals and solving the HF RoothaanHall equations. A medium-sized basis set will typically have functions for each atom, and already a atom system may result in several thousand basis functions, and thus potentially , two-electron integrals.
In a traditionalimplementation, these integrals are calculated and stored on disk, requiring ,10 TB of disk space. A straightforward implementation of the HF model is,therefore, an N 4 method, increasing the system size by a factor of two will increase the computational time and storage by a factor of 16, and this effectively limits the application to systems with less than ,30 atoms. Two developments have been essential for reducing the scaling and thereby pushing the limit for feasible calculations. The first is the introduction of the direct SCFmethod with differential update of the Fock matrix .
The Fock matrix can be written as a one-electron contribution h, which is independent of the density matrix, and a contraction of the density matrix D with a two-electron tensor G. The change in the Fock matrix is thus given by the change in the density matrix. Fi h DiG F. Jensen6 DFi DDiG Rather than calculating all the two-electron integrals prior to solving the RoothaanHall equation, the integrals can be recalculated in each iteration. While this avoids the requirement for massive amounts of disk storage, it potentially increases the computational timewith a factor close to the number of iterations.
The availability of the density matrix elements, however, means that not all integrals have to be calculated, only those that will bemultiplied with sufficiently large density matrix elements are required. The density matrix elements between atoms that are spatially far apart will be close to zero and this effectively reduces the method scaling from N4 to N2 for large systems.
Furthermore, as the iterative solution proceeds, the change in thedensitymatrix hopefully becomessmaller andsmaller and the integral screening therefore becomes more and more efficient. For large systems, the dominating integrals are those describing the Coulomb interaction between electrons, leading to an overall N2 scaling. In fast-multipole methods, the Coulomb contribution is not calculated by two-electron integrals, but is replaced with the interaction between two electron densities .
The latter can be calculated in a more efficient fashion by partitioning the physical space into boxes and evaluating the interaction between densities within the boxes as interactions between multipoles located at the center of the boxes. The required multipole order and box size depends on the distance between boxes for a given final accuracy, and distant interactions can, therefore, be calculated with a coarser granulation than the near-field contribution.
For sufficiently large systems, this leads to a computational complexity of order N, i. Solution of the RoothaanHall equation by repeated diagonalization of the Fock matrix requires a computational time proportional to the cube of the matrix dimension. Imprint: Elsevier. Published Date: 25th November Page Count: View all volumes in this series: Annual Reports in Computational Chemistry.
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Institutional Subscription. Free Shipping Free global shipping No minimum order. Introduction 2. Theory 3. Summary and Outlook Chapter Two. Roadmap: Future Directions and Challenges 6. Conclusions Chapter Three. Theoretical Background 3. Final Remarks Chapter Four. Chemical Bonding and Symmetry of B80 3. Reactivity and Aromaticity of B80 4.