Full-CI quantum chemistry using the density matrix renormalization group

Abstract

We describe how density-matrix renormalization group (DMRG) can be used to solve the full-CI problem in quantum chemistry. As an illustration of the potential of this method, we apply it to a paramagnetic molecule. In particular, we show the effect of various basis set, the scaling as the fourth power of the size of the problem, and compare the DMRG with other methods. I. INTRODUCTIONFirst-principle quantum chemistry is employed successfully to obtain thermochemical data; molecular structures; force fields and frequencies; assignments of NMR-, photoelectron-, E.S.R-, and UV-spectra; transition state structures as well as activation barriers; dipole moments and other one-or two-electron properties. Two routes of calculation are available: (i) ab initio Hartree-Fock (HF) and post-HF methods; and (ii) density functional theory (DFT). Though both approaches are rigorous, the former one necessitates lengthy configuration interaction (CI) treatment to account for electron correlation; whereas the latter one crucially depends upon the quest for accurate exchange and correlation functionals. The recently acquired popularity of DFT stems in large measure from its computational efficiency, allowing it to treat medium to large size molecules at a fraction of the time required for HF or post-HF calculations. More importantly, expectation values derived from DFT are, in most cases, better in line with experiment than results obtained from HF calculations. This is particularly the case for systems involving transition metals. Nevertheless, if one wants to achieve experimental accuracy for small polyatomic molecules, the method reaches its limits. On the other hand, post-HF overcomes these limits and goes even beyond experimental accuracy, e.g. in case of H 2 . The drawback is of course its very high cost in computational power. However these very accurate calculations of small systems may provide the best route to obtaining more accurate exchange-correlation potentials using the constrained search algorithm of Levy [1].Keeping the discussion to ab initio (HF and post-HF) quantum chemistry, let us now consider the state of the art in this topic. We now know how to do very large calculations, using the direct methodology [2]. We can also manage to work with good basis sets for such calculations, although it is considered that 6-31G* are not good enough, and probably something nearer to TZ2P is required for definitive SCF calculations. Beyond SCF there are major difficulties, all associated with trying to more accurately represent the electron-electron cusp. We know from the work of Kutzelnigg [3], that the convergence of this problem is very slow, something like (ℓ + 1 2 ) −4 , with ℓ the orbital angular momentum quantum number. This means that very large basis sets are required for correlated calculations. We know that it is more important to include d and f basis functions than to improve the methodology (e.g. 6-31G* basis are not appropriate for correlated studies). We also know that the raw cost of the main correlated method...

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