Geometry optimization, Binding of molecules

Title:

Geometry optimization, Binding of molecules

Name:

Gero Friesecke, Florian Theil

Affil./Addr. 1:

Center of Mathematics, TU Munich [email protected]

Affil./Addr. 2:

Mathematics Institute, University of Warwick [email protected]

Geometry optimization, Binding of molecules Mathematics Subject classification 81V55, 70Cxx, 92C40

Short Definition Geometry optimization is a method to predict the three-dimensional arrangement of the atoms in a molecule by means of minimization of a model energy. The phenomenon of binding, that is to say the tendency of atoms and molecules to conglomerate into stable larger structures, as well as the emergence of specific structures depending on the constituting elements, can be explained, at least in principle, as a result of geometry optimization.

Pheonomena Two atoms are said to be linked together by a bond if there is an opposing force against pulling them apart. Associated with a bond is a binding energy, which is the total energy required to separate the atoms. Except at very high temperature, atoms form bonds between each other and conglomerate into molecules and larger aggregates such as atomic or molecular chains, clusters, and crystals. The ensuing molecular geometries, that is to say the 3D arrangements of the atoms, and the binding energies of the different bonds, crucially influence physical and chemical behaviour. Therefore, theoretically predicting them forms a large and important part of contemporary research in chemistry, materials science, and molecular biology. A major difficulty is that binding energies, preferred partners, and local geometries are highly chemically specific, that is to say they depend on the elements involved. For instance, the experimental binding energies of the diatomic molecules Li2 , Be2 , and N2 (i.e. the dimers of element number 3, 4, 7 in the periodic table) are roughly in the ratio 10 : 1 : 100. And CH2 is bent, whereas CO2 is straight. When atoms form bonds, their electronic structure, that is to say the probability cloud of electrons around their atomic nucleus, re-arranges. Chemists distinguish phenomenologically between different types of bonds, depending on this type of re-arrangement: covalent, ionic, and metallic bonds, as well as weak bonds such as hydrogen- or van-derWaals-bonds. A covalent bond corresponds to a substantial re-arrangement of the electron cloud into the space between the atoms while each atom maintains a net charge neutrality, as in the C–C bond. In a ionic bond, one electron migrates almost fully to the other atom,

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as in the dimer Na–Cl. The metallic bond between atoms in a solid metal is pictured as the formation of a “sea” of free electrons, no longer associated to any particular atom, surrounding a lattice of ionic cores. The above distinctions, albeit a helpful guide, should not be taken too literally, and are often not so clear-cut in practice. A unifying theoretical viewpoint of the 3D molecular structures resulting from interatomic bonding, regardless of the type of bonds, is to view them as geometry optimizers, i.e. as locally or globally optimal spatial arrangements of the atoms which minimize overall energy. For a mathematical formulation see Section 2. If the number of atoms or molecules is large (& 100), then the system will start behaving in a thermodynamic way. At sufficiently low temperature, identical atoms or molecules typically arrange themselves into a crystal, that is to say the positions of the atomic nuclei are given approximately by a subset of a crystal lattice. A crystal lattice L is a finite union of discrete subsets of R3 of form {ie + jf + kg | i, j, k ∈ Z}, where e, f, g are linearly independent vectors in R3 . Near the boundaries of crystals, the underlying lattice is often distorted. Closely related effects are the emergence of defects such as vacancies, interstitial atoms, dislocations and continuum deformations. Vacancies and interstitial atoms are missing respectively additional atoms. Dislocations are topological crystallographic defects which can sometimes be visualized as being caused by the termination of a plane of atoms in the middle of a crystal. Continuum deformations are small long-wavelength distortions of the underlying lattice arising from external loads, as in an elastically bent macroscopic piece of metal. A unifying interpretation of the above structures arises by extending the term ‘geometry optimization’, which is typically used in connection with single molecules, to large scale 3

systems as well. The spatial arrangemens of the atoms can again be understood, at least locally and subject to holding the atomic positions in an outer region fixed, as geometry optimizers, i.e. minimizers of energy.

Geometry optimization and binding energy prediction Geometry optimization, in its basic all-atom form, makes a prediction for the 3D spatial arrangement of the atoms in a molecule, by a two-step procedure. Suppose the system consists of M atoms, with atomic numbers Z1 , .., ZM . Step A: Specify a model energy, or potential energy surface (PES), that is to say a function Φ : R3M → R ∪ {+∞} which gives the system’s potential energy as a function of the vector X = (X1 , .., XM ) ∈ R3M of the atomic positions Xj ∈ R3 . Step B: Compute (local or global) minimizers (X1 , .., XM ) of Φ. Basic physical quantities of the molecule correspond to mathematical quantities of the energy surface as follows: binding energy

difference between minimum energy and sum of energies of subsystems

stable configuration local minimizer transition state

saddle point

bond length/angle parameter in minimizing configuration

More precisely, the theoretical binding energy ∆E of the minimizer obtained in Step B with respect to decomposition into two subsystems, say of the first K atoms and the last M − K atoms, is defined as ∆E = min Φ(X) − lim min{Φ(X) : dist({X1 , . . . , XK }, {XK+1 , . . . , XM }) ≥ R}. R→∞

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Potential energy surfaces have the general property of Galileian invariance, that is to say Φ(X1 , .., XM ) = Φ(RX1 + a, .., RXM + a), for any translation vector a ∈ R3 and any rotation matrix R ∈ SO(3). Thus a one-atom surface Φ(X1 ) is independent of X1 , and a two-atom surface Φ(X1 , X2 ) equals ϕ(|X1 − X2 |) for some function of interatomic distance. In particular, for a diatomic molecule, the geometry optimization step B reduces to computing the bond length, r∗ := argminr ϕ(r).

Model energies A wide range of model energies are in use, depending on the type of system and the desired level of understanding. To obtain quantitatively accurate and chemically specific predictions, one uses ab initio energy surfaces, that is to say surfaces obtained from a quantum mechanical model for the system’s electronic structure which requires as input only atomic numbers. For large systems, one often uses classical potentials. The latter are particularly useful for predicting the 3D structure of systems composed from many identical copies of just a few basic units, such as crystalline clusters, carbon nanotubes, or nucleic acids.

Born-Oppenheimer potential energy surface The gold standard model energy of a system of M atoms, which in principle contains the whole range of phenomena described in Section 1, is the ground state Born-Oppenheimer PES of non-relativistic quantum mechanics. With X = (X1 , .., XM ) ∈ R3M denoting the vector of nuclear positions, it has the general mathematical form ΦBO (X) = min E(X, Ψ ),

(1)

Ψ ∈AN

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where E is an energy functional depending on an infinite-dimensional field Ψ , the electronic wavefunction. For a molecule with N electrons, the latter is a function on the configuration space (R3 × Z2 )N of the electron positions and spins. More precisely AN = {Ψ ∈ L2 ((R3 × Z2 )N ) → C | ||Ψ ||L2 = 1, ∇Ψ ∈ L2 , Ψ antisymmetric}, where antisymmetric means, with xi , si denoting the position and spin of the ith electron, Ψ (..., xi , si , ..., xj , sj , ...) = −Ψ (..., xj , sj , ..., xi , si , ...) for all i < j. The functional E is given, in atomic units, by E(X, Ψ ) = H = vX (x1 ) +

N X

∇2xi +

R (R3 ×Z2 )N

X

Ψ ∗ HΨ where

Wee (xi − xj ) + Wnn (X)

(2)

X Zα Zβ 1 Zα , Wee (r) = and Wnn (X) = , |r − Xα | |r| |X α − Xβ | 1≤α