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Introduction to First-Principle Simulation of Molecular Systems

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Computational Mathematics, Numerical Analysis and Applications

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Abstract

First-principle molecular simulation aims at computing the physical and chemical properties of a molecule, or more generally of a material system, from the fundamental laws of quantum mechanics. It is widely used in various application fields ranging from quantum chemistry to materials science and molecular biology, and is the source of many very interesting and challenging mathematical and numerical problems. This chapter is an elementary introduction to this field, covering some modeling, mathematical, and numerical aspects.

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Notes

  1. 1.

    Again by Riesz representation theorem.

  2. 2.

    Recall that a matrix \(A \in \mathbb{C}^{d\times d}\) is called Hermitian if A ∗ = A (i.e. \(\overline{A_{ij}} = A_{ji}\), \(\forall 1 \leq i,j \leq d\)). If \(z \in \mathbb{C}\) and \(A \in \mathbb{C}^{d\times d}\), we use the shorthand notation z − A to denote the matrix zI d − A, where I d is the rank-d identity matrix. We proceed similarly with linear operators on complex Hilbert spaces.

  3. 3.

    The operator A is called closed if its graph \(\varGamma (A):= \left \{(u,Au),\,u \in D(A)\right \}\) is a closed subspace of \(\mathscr{H} \times \mathscr{ H}\).

  4. 4.

    Since A is bounded below, there exists \(C \in \mathbb{R}\) s.t. (u, v) Q(A): = 〈u | Av〉 + C〈u | v〉 is a scalar product on D(A). The Cauchy closure of D(A) for the associated norm is a Hilbert space, independent of C, called the form domain of A. The quadratic form associated with A is the unique continuous extension of (u, v) ↦ 〈u | Av〉 to Q(A).

  5. 5.

    We limit ourselves to pure states in these lectures notes.

  6. 6.

    It may seem weird that steady states explicitly depend on time. This apparent paradox is due to the fact that a state is in fact an element of the projective space \((\mathscr{H}\setminus \left \{0\right \})/\mathbb{C}^{{\ast}}\), so that f(t)ψ and ψ actually represent the exact same state.

  7. 7.

    An operator A on \(L^{2}(\mathbb{R}^{d})\) such that ρ(A) ≠ ∅ is called locally compact if for any bounded set B, the operator χ B (z − A)−1 is a compact operator on \(L^{2}(\mathbb{R}^{d})\) for some (and then all by virtue of the resolvent formula) z ∈ ρ(A). Here, χ B is the characteristic function of B; in the expression χ B (z − A)−1 , χ B should be understood as the multiplication operator by the bounded function χ B , which is a bounded self-adjoint operator on \(\mathscr{H}\). The Hamiltonian of the hydrogen atom is a locally compact self-adjoint operator on \(L^{2}(\mathbb{R}^{3})\), and for this operator, \(\mathit{\mbox{ dim}}(\mathscr{H}_{p}) = \mathit{\mbox{ dim}}(\mathscr{H}_{c}) = \infty\).

  8. 8.

    For simplicity, we omit the spin variables. See Remark 25 below for more details.

  9. 9.

    These are the most common isotopes of oxygen and hydrogen.

  10. 10.

    Eigenvalues are counted with their multiplicities, so that \(E_{0}^{\left \{\mathbf{R}_{k}\right \}} \leq E_{1}^{\left \{\mathbf{R}_{k}\right \}} \leq E_{2}^{\left \{\mathbf{R}_{k}\right \}}\cdots\), with a priori large inequalities.

  11. 11.

    Breakdowns of the adiabatic approximation are studied in [14, 22, 35].

  12. 12.

    The operator A has a compact resolvent if, for some z ∈ ρ(A) (and therefore for all z ∈ ρ(A) by virtue of the resolvent formula), z − A, considered as a bounded operator on \(\mathscr{H}\), is compact.

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Acknowledgements

I am grateful to the organizers of the XVII Jacques-Louis Lions Spanish-French School on Numerical Simulation in Physics and Engineering for inviting me to deliver a course. Special thanks to Mariano Mateos Alberdi for the great local organization.

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  1. CERMICS, Ecole des Ponts and Inria Paris, 6 & 8 avenue Blaise Pascal, 77455, Marne-la-Vallée, France

    Eric Cancès

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  1. Matemáticas, Universidad de Oviedo , Gijón, Asturias, Spain

    Mariano Mateos

  2. Matemáticas, Universidad de Oviedo , Gijón, Asturias, Spain

    Pedro Alonso

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Cancès, E. (2017). Introduction to First-Principle Simulation of Molecular Systems. In: Mateos, M., Alonso, P. (eds) Computational Mathematics, Numerical Analysis and Applications. SEMA SIMAI Springer Series, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-49631-3_2

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