Linear scaling method, Hartree-Fock methods

Hartree-Fock Methods for Large Systems Linear Scaling Methods... 

Kussmann, J., 8c Ochsenfeld, C. (2007). Linear-scaling method for calculating nuclear magnetic resonance chemical shifts using gauge-including atomic orbitals within Hartree-Fock and density-functional theory. Journal of Chemical Physics, 127, 054103. 

Basis Sets Correlation Consistent Sets Configuration Interaction Coupled-cluster Theory Density Functional Applications Density Functional Theory Applications to Transition Metal Problems G2 Theory Integrals of Electron Repulsion Integrals Overlap Linear Scaling Methods for Electronic Structure Calculations Localized MO SCF Methods Mpller-Plesset Perturbation Theory Monte Carlo Quantum Methods for Electronic Structure Numerical Hartree-Fock Methods for Molecules Pseudospectral Methods in Ab Initio Quantum Chemistry Self-consistent Reaction Field Methods Symmetry in Hartree-Fock Theory. 

The Hartree-Fock method can be applied to truly large molecules containing several hundred atoms. For such systems, it becomes impossible to construct a set of orthonormal orbitals (5.1.5), much less a set of canonical orbitals. However, as we shall see in Chapter 10, all information about the Hartree-Fock wave function is contained in the one-electron density matrix, which may be expressed directly in the basis of AOs. For large molecules, the density-matrix elements can be optimized by an algorithm whose complexity scales linearly with the size of the system. 

The problem with most quantum mechanical methods is that they scale badly. This means that, for instance, a calculation for twice as large a molecule does not require twice as much computer time and resources (this would be linear scaling), but rather 2" times as much, where n varies between about 3 for DFT calculations to 4 for Hartree-Fock and very large numbers for ab-initio techniques with explicit treatment of electron correlation. Thus, the size of the molecules that we can treat with conventional methods is limited. Linear scaling methods have been developed for ab-initio, DFT and semi-empirical methods, but only the latter are currently able to treat complete enzymes. There are two different approaches available. 

The concept of purification is well known in the linear-scaling literature for one-particle theories like Hartree-Fock and density functional theory, where it denotes the iterative process by which an arbitrary one-particle density matrix is projected onto an idempotent 1-RDM [2,59-61]. An RDM is said to be pure A-representable if it arises from the integration of an Al-particle density matrix T T, where T (the preimage) is an Al-particle wavefiinction [3-5]. Any idempotent 1-RDM is N-representable with a unique Slater-determinant preimage. Within the linear-scaling literature the 1-RDM may be directly computed with unconstrained optimization, where iterative purification imposes the A-representabUity conditions [59-61]. Recently, we have shown that these methods for computing the 1 -RDM directly... 

In this section, we briefly discuss some of the electronic structure methods which have been used in the calculations of the PE functions which are discussed in the following sections. There are variety of ab initio electronic structure methods which can be used for the calculation of the PE surface of the electronic ground state. Most widely used are Hartree-Fock (HF) based methods. In this approach, the electronic wavefunction of a closed-shell system is described by a determinant composed of restricted one-electron spin orbitals. The unrestricted HF (UHF) method can handle also open-shell electronic systems. The limitation of HF based methods is that they do not account for electron correlation effects. For the electronic ground state of closed-shell systems, electron correlation effects can be accounted for relatively easily by second-order Mpller-Plesset perturbation theory (MP2). In modern implementations of MP2, linear scaling with the size of the system has been achieved. It is thus possible to treat quite large molecules and clusters at this level of theory. 

Usually, the parent molecules M are confined to some limited size that allows rapid determination of the parent molecule density matrices within a conventional ab initio Hartree-Fock-Roothaan-Hall scheme, followed by the determination of the fragment density matrices and the assembly of the macro-molecular density matrix using the method described above. The entire iterative procedure depends linearly on the number of fragments, that is, on the size of the target macromolecule M. When compared to the conventional ab initio type methods of computer time requirements growing with the third or fourth power of the number of electrons, the linear scaling property of the ADMA method is advantageous. 

Q-Chem (www.q-chem.com/), first released in 1997, is an ab initio package that allows calculations on large molecules (several hundred atoms) and can do Hartree-Fock, MP2, and density-functional calculations. It incorporates methods such as CFMM and ONX to achieve linear scaling (Section 15.5) for large molecules. 

Experimental chemistry is focused, in most cases, on molecules of a larger size than those for which fair calculations with correlation are possible. However, after thorough analysis of the situation, it turns out that the cost of the calculations does not necessarily increase very fast with the size of a molecule. Employing localized molecular orbitals and using the multipole expansion (see Appendix X available at booksite.elsevier.com/978-0-444-59436-5) of the integrals involving the orbitals separated in space causes, fa- elongated molecules, the cost of the post-Hartree-Fock calculations to scale linearly with the size of a molecule. It can he expected that if the methods described in... 

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