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The Schrödinger equation

the Schrödinger equation

$H(t)\left|\psi \left(t\right)\right\rangle =\mathrm {i} \hbar {\frac {\partial }{\partial t}}\left|\psi \left(t\right)\right\rangle$

where $\mathrm {i}$ is the imaginary unit, $t$ is time, $\partial /\partial t$ is the partial derivative with respect to $t$, $\hbar$ is the reduced Planck's constant (Planck's constant divided by $2\pi$), $\psi (t)$ is the wave function, and $H\!\left(t\right)$ is the Hamiltonian operator.

Hartree-Fock (HF) or self-consistent field (SCF)

In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.

The Hartree–Fock method often assumes that the exact, N-body wave function of the system can be approximated by a single Slater determinant (in the case where the particles are fermions) or by a single permanent (in the case of bosons) of N spin-orbitals. By invoking the w:variational method, one can derive a set of N-coupled equations for the N spin orbitals. A solution of these equations yields the Hartree–Fock wave function and energy of the system.

Especially in the older literature, the Hartree–Fock method is also called the self-consistent field method (SCF). In deriving what is now called the Hartree equation as an approximate solution of the Schrödinger equation, Hartree required the final field as computed from the charge distribution to be "self-consistent" with the assumed initial field. Thus, self-consistency was a requirement of the solution. The solutions to the non-linear Hartree–Fock equations also behave as if each particle is subjected to the mean field created by all other particles (see the Fock operator below) and hence the terminology continued. The equations are almost universally solved by means of an iterative method, although the fixed-point iteration algorithm does not always converge. This solution scheme is not the only one possible and is not an essential feature of the Hartree–Fock method.

The Hartree–Fock method finds its typical application in the solution of the Schrödinger equation for atoms, molecules, nanostructures and solids but it has also found widespread use in nuclear physics. (See Hartree–Fock–Bogoliubov method for a discussion of its application in nuclear structure theory). In atomic structure theory, calculations may be for a spectrum with many excited energy levels and consequently the Hartree–Fock method for atoms assumes the wave function is a single configuration state function with well-defined quantum numbers and that the energy level is not necessarily the ground state.

For both atoms and molecules, the Hartree–Fock solution is the central starting point for most methods that describe the many-electron system more accurately.

The rest of this article will focus on applications in electronic structure theory suitable for molecules with the atom as a special case. The discussion here is only for the Restricted Hartree–Fock method, where the atom or molecule is a closed-shell system with all orbitals (atomic or molecular) doubly occupied. Open-shell systems, where some of the electrons are not paired, can be dealt with by one of two Hartree–Fock methods:

• w:Restricted open-shell Hartree–Fock (ROHF)
• w:Unrestricted Hartree–Fock (UHF)

history

The origin of the Hartree–Fock method dates back to the end of the 1920s, soon after the discovery of the w:Schrödinger equation in 1926. In 1927 D. R. Hartree introduced a procedure, which he called the self-consistent field method, to calculate approximate wave functions and energies for atoms and ions. Hartree was guided by some earlier, semi-empirical methods of the early 1920s (by E. Fues, R. B. Lindsay, and himself) set in the w:old quantum theory of Bohr.

In the w:Bohr model of the atom, the energy of a state with w:principal quantum number n is given in atomic units as $E=-1/n^{2}$. It was observed from atomic spectra that the energy levels of many-electron atoms are well described by applying a modified version of Bohr's formula. By introducing the w:quantum defect d as an empirical parameter, the energy levels of a generic atom were well approximated by the formula $E=-1/(n+d)^{2}$, in the sense that one could reproduce fairly well the observed transitions levels observed in the w:X-ray region (for example, see the empirical discussion and derivation in w:Moseley's law). The existence of a non-zero quantum defect was attributed to electron-electron repulsion, which clearly does not exist in the isolated hydrogen atom. This repulsion resulted in partial screening of the bare nuclear charge. These early researchers later introduced other potentials containing additional empirical parameters with the hope of better reproducing the experimental data.

Hartree sought to do away with empirical parameters and solve the many-body time-independent Schrödinger equation from fundamental physical principles, i.e., ab initio. His first proposed method of solution became known as the Hartree method. However, many of Hartree's contemporaries did not understand the physical reasoning behind the Hartree method: it appeared to many people to contain empirical elements, and its connection to the solution of the many-body Schrödinger equation was unclear. However, in 1928 J. C. Slater and J. A. Gaunt independently showed that the Hartree method could be couched on a sounder theoretical basis by applying the w:variational principle to an w:ansatz (trial wave function) as a product of single-particle functions.

In 1930 Slater and V. A. Fock independently pointed out that the Hartree method did not respect the principle of antisymmetry of the wave function. The Hartree method used the w:Pauli exclusion principle in its older formulation, forbidding the presence of two electrons in the same quantum state. However, this was shown to be fundamentally incomplete in its neglect of w:quantum statistics.

It was then shown that a w:Slater determinant, a w:determinant of one-particle orbitals first used by Heisenberg and Dirac in 1926, trivially satisfies the antisymmetric property of the exact solution and hence is a suitable w:ansatz for applying the w:variational principle. The original Hartree method can then be viewed as an approximation to the Hartree–Fock method by neglecting exchange. Fock's original method relied heavily on w:group theory and was too abstract for contemporary physicists to understand and implement. In 1935 Hartree reformulated the method more suitably for the purposes of calculation.

The Hartree–Fock method, despite its physically more accurate picture, was little used until the advent of electronic computers in the 1950s due to the much greater computational demands over the early Hartree method and empirical models. Initially, both the Hartree method and the Hartree–Fock method were applied exclusively to atoms, where the spherical symmetry of the system allowed one to greatly simplify the problem. These approximate methods were (and are) often used together with the w:central field approximation, to impose that electrons in the same shell have the same radial part, and to restrict the variational solution to be a spin eigenfunction. Even so, solution by hand of the Hartree–Fock equations for a medium sized atom were laborious; small molecules required computational resources far beyond what was available before 1950.

Hartree–Fock algorithm

The Hartree–Fock method is typically used to solve the time-independent Schrödinger equation for a multi-electron atom or molecule as described in the w:Born–Oppenheimer approximation. Since there are no known solutions for many-electron systems (hydrogenic atoms and the diatomic hydrogen cation being notable one-electron exceptions), the problem is solved numerically. Due to the nonlinearities introduced by the Hartree–Fock approximation, the equations are solved using a nonlinear method such as w:iteration, which gives rise to the name "self-consistent field method."

Approximations

The Hartree–Fock method makes five major simplifications in order to deal with this task:

• The w:Born–Oppenheimer approximation is inherently assumed. The full molecular wave function is actually a function of the coordinates of each of the nuclei, in addition to those of the electrons.
• Typically, relativistic effects are completely neglected. The momentum operator is assumed to be completely non-relativistic.
• The variational solution is assumed to be a w:linear combination of a finite number of basis functions, which are usually (but not always) chosen to be w:orthogonal. The finite basis set is assumed to be approximately complete.
• Each w:energy eigenfunction is assumed to be describable by a single w:Slater determinant, an antisymmetrized product of one-electron wave functions (i.e., orbitals).
• The mean field approximation is implied. Effects arising from deviations from this assumption, known as w:electron correlation, are completely neglected for the electrons of opposite spin, but are taken into account for electrons of parallel spin. (Electron correlation should not be confused with electron exchange, which is fully accounted for in the Hartree–Fock method.)

Relaxation of the last two approximations give rise to many so-called w:post-Hartree–Fock methods.

Greatly simplified algorithmic flowchart illustrating the Hartree–Fock method

Variational optimization of orbitals

The variational theorem states that for a time-independent Hamiltonian operator, any trial wave function will have an energy w:expectation value that is greater than or equal to the true w:ground state wave function corresponding to the given Hamiltonian. Because of this, the Hartree–Fock energy is an upper bound to the true ground state energy of a given molecule. In the context of the Hartree–Fock method, the best possible solution is at the Hartree–Fock limit; i.e., the limit of the Hartree–Fock energy as the basis set approaches completeness. (The other is the full-CI limit, where the last two approximations of the Hartree–Fock theory as described above are completely undone. It is only when both limits are attained that the exact solution, up to the Born–Oppenheimer approximation, is obtained.) The Hartree–Fock energy is the minimal energy for a single Slater determinant.

The starting point for the Hartree–Fock method is a set of approximate one-electron wave functions known as w:spin-orbitals. For an w:atomic orbital calculation, these are typically the orbitals for a hydrogenic atom (an atom with only one electron, but the appropriate nuclear charge). For a w:molecular orbital or crystalline calculation, the initial approximate one-electron wave functions are typically a w:linear combination of atomic orbitals (LCAO).

The orbitals above only account for the presence of other electrons in an average manner. In the Hartree–Fock method, the effect of other electrons are accounted for in a w:mean-field theory context. The orbitals are optimized by requiring them to minimize the energy of the respective Slater determinant. The resultant variational conditions on the orbitals lead to a new one-electron operator, the w:Fock operator. At the minimum, the occupied orbitals are eigensolutions to the Fock operator via a w:unitary transformation between themselves. The Fock operator is an effective one-electron Hamiltonian operator being the sum of two terms. The first is a sum of kinetic energy operators for each electron, the internuclear repulsion energy, and a sum of nuclear-electronic Coulombic attraction terms. The second are Coulombic repulsion terms between electrons in a mean-field theory description; a net repulsion energy for each electron in the system, which is calculated by treating all of the other electrons within the molecule as a smooth distribution of negative charge. This is the major simplification inherent in the Hartree–Fock method, and is equivalent to the fifth simplification in the above list.

Since the Fock operator depends on the orbitals used to construct the corresponding w:Fock matrix, the eigenfunctions of the Fock operator are in turn new orbitals which can be used to construct a new Fock operator. In this way, the Hartree–Fock orbitals are optimized iteratively until the change in total electronic energy falls below a predefined threshold. In this way, a set of self-consistent one-electron orbitals are calculated. The Hartree–Fock electronic wave function is then the Slater determinant constructed out of these orbitals. Following the basic postulates of quantum mechanics, the Hartree–Fock wave function can then be used to compute any desired chemical or physical property within the framework of the Hartree–Fock method and the approximations employed.

Mathematical formulation

The Fock operator

Main article: w:Fock matrix

Because the electron-electron repulsion term of the w:electronic molecular Hamiltonian involves the coordinates of two different electrons, it is necessary to reformulate it in an approximate way. Under this approximation, (outlined under Hartree–Fock algorithm), all of the terms of the exact Hamiltonian except the nuclear-nuclear repulsion term are re-expressed as the sum of one-electron operators outlined below, for closed-shell atoms or molecules (with two electrons in each spatial orbital). The "(1)" following each operator symbol simply indicates that the operator is 1-electron in nature.

${\hat {F}}[\{\phi _{j}\}](1)={\hat {H}}^{\text{core}}(1)+\sum _{j=1}^{N/2}[2{\hat {J}}_{j}(1)-{\hat {K}}_{j}(1)]$

where

${\hat {F}}[\{\phi _{j}\}](1)$

is the one-electron Fock operator generated by the orbitals $\phi _{j}$, and

${\hat {H}}^{\text{core}}(1)=-{\frac {1}{2}}\nabla _{1}^{2}-\sum _{\alpha }{\frac {Z_{\alpha }}{r_{1\alpha }}}$

is the one-electron core Hamiltonian. Also

${\hat {J}}_{j}(1)$

is the w:Coulomb operator, defining the electron-electron repulsion energy due to each of the two electrons in the jth orbital. Finally

${\hat {K}}_{j}(1)$

is the w:exchange operator, defining the electron exchange energy due to the antisymmetry of the total n-electron wave function. This (so called) "exchange energy" operator, K, is simply an artifact of the Slater determinant. Finding the Hartree–Fock one-electron wave functions is now equivalent to solving the eigenfunction equation:

${\hat {F}}(1)\phi _{i}(1)=\epsilon _{i}\phi _{i}(1)$

where $\phi _{i}\;(1)$ are a set of one-electron wave functions, called the Hartree–Fock molecular orbitals.

Linear combination of atomic orbitals

Main articles: w:basis set (chemistry) and w:basis set

Typically, in modern Hartree–Fock calculations, the one-electron wave functions are approximated by a w:linear combination of atomic orbitals. These atomic orbitals are called w:Slater-type orbitals. Furthermore, it is very common for the "atomic orbitals" in use to actually be composed of a linear combination of one or more Gaussian-type orbitals, rather than Slater-type orbitals, in the interests of saving large amounts of computation time.

Various basis sets are used in practice, most of which are composed of Gaussian functions. In some applications, an orthogonalization method such as the w:Gram–Schmidt process is performed in order to produce a set of orthogonal basis functions. This can in principle save computational time when the computer is solving the Roothaan–Hall equations by converting the w:overlap matrix effectively to an w:identity matrix. However, in most modern computer programs for molecular Hartree–Fock calculations this procedure is not followed due to the high numerical cost of orthogonalization and the advent of more efficient, often sparse, algorithms for solving the w:generalized eigenvalue problem, of which the Roothaan–Hall equations are an example.

Numerical stability

w:Numerical stability can be a problem with this procedure and there are various ways of combating this instability. One of the most basic and generally applicable is called F-mixing or damping. With F-mixing, once a single electron wave function is calculated it is not used directly. Instead, some combination of that calculated wave function and the previous wave functions for that electron is used—the most common being a simple linear combination of the calculated and immediately preceding wave function. A clever dodge, employed by Hartree, for atomic calculations was to increase the nuclear charge, thus pulling all the electrons closer together. As the system stabilised, this was gradually reduced to the correct charge. In molecular calculations a similar approach is sometimes used by first calculating the wave function for a positive ion and then to use these orbitals as the starting point for the neutral molecule. Modern molecular Hartree–Fock computer programs use a variety of methods to ensure convergence of the Roothaan–Hall equations.

Weaknesses, extensions, and alternatives

Of the five simplifications outlined in the section "Hartree–Fock algorithm", the fifth is typically the most important. Neglecting electron correlation can lead to large deviations from experimental results. A number of approaches to this weakness, collectively called w:post-Hartree–Fock methods, have been devised to include electron correlation to the multi-electron wave function. One of these approaches, w:Møller–Plesset perturbation theory, treats correlation as a perturbation of the Fock operator. Others expand the true multi-electron wave function in terms of a linear combination of Slater determinants—such as w:multi-configurational self-consistent field, w:configuration interaction, w:quadratic configuration interaction, and complete active space SCF (CASSCF). Still others (such as variational quantum Monte Carlo) modify the Hartree–Fock wave function by multiplying it by a correlation function ("Jastrow" factor), a term which is explicitly a function of multiple electrons that cannot be decomposed into independent single-particle functions.

An alternative to Hartree–Fock calculations used in some cases is w:density functional theory, which treats both exchange and correlation energies, albeit approximately. Indeed, it is common to use calculations that are a hybrid of the two methods—the popular B3LYP scheme is one such w:hybrid functional method. Another option is to use w:modern valence bond methods.

Software packages

For a list of software packages known to handle Hartree–Fock calculations, particularly for molecules and solids, see the w:list of quantum chemistry and solid state physics software.

Sources

• Levine, Ira N. (1991). Quantum Chemistry (4th ed.). Englewood Cliffs, New Jersey: Prentice Hall. pp. 455–544. ISBN 0-205-12770-3.
• Cramer, Christopher J. (2002). Essentials of Computational Chemistry. Chichester: John Wiley & Sons, Ltd. pp. 153–189. ISBN 0-471-48552-7.
• Szabo, A.; Ostlund, N. S. (1996). Modern Quantum Chemistry. Mineola, New York: Dover Publishing. ISBN 0-486-69186-1.

Slater determinant

Two-particle case

The simplest way to approximate the wave function of a many-particle system is to take the product of properly chosen orthogonal wave functions of the individual particles. For the two-particle case with spatial coordinates $\mathbf {x} _{1}$ and $\mathbf {x} _{2}$, we have

$\Psi (\mathbf {x} _{1},\mathbf {x} _{2})=\chi _{1}(\mathbf {x} _{1})\chi _{2}(\mathbf {x} _{2}).$

This expression is used in the w:Hartree–Fock method as an w:ansatz for the many-particle wave function and is known as a w:Hartree product. However, it is not satisfactory for w:fermions because the wave function above is not antisymmetric, as it must be for w:fermions from the w:Pauli exclusion principle. An antisymmetric wave function can be mathematically described as follows:

$\Psi (\mathbf {x} _{1},\mathbf {x} _{2})=-\Psi (\mathbf {x} _{2},\mathbf {x} _{1})$

which does not hold for the Hartree product. Therefore the Hartree product does not satisfy the Pauli principle. This problem can be overcome by taking a w:linear combination of both Hartree products

$\Psi (\mathbf {x} _{1},\mathbf {x} _{2})={\frac {1}{\sqrt {2}}}\{\chi _{1}(\mathbf {x} _{1})\chi _{2}(\mathbf {x} _{2})-\chi _{1}(\mathbf {x} _{2})\chi _{2}(\mathbf {x} _{1})\}$

$={\frac {1}{\sqrt {2}}}{\begin{vmatrix}\chi _{1}(\mathbf {x} _{1})&\chi _{2}(\mathbf {x} _{1})\\\chi _{1}(\mathbf {x} _{2})&\chi _{2}(\mathbf {x} _{2})\end{vmatrix}}$

where the coefficient is the w:normalization factor. This wave function is now antisymmetric and no longer distinguishes between fermions, that is: one cannot indicate an ordinal number to a specific particle and the indices given are interchangeable. Moreover, it also goes to zero if any two wave functions of two fermions are the same. This is equivalent to satisfying the Pauli exclusion principle.

Generalizations

The expression can be generalised to any number of fermions by writing it as a w:determinant. For an N-electron system, the Slater determinant is defined as

$\Psi (\mathbf {x} _{1},\mathbf {x} _{2},\ldots ,\mathbf {x} _{N})={\frac {1}{\sqrt {N!}}}\left|{\begin{matrix}\chi _{1}(\mathbf {x} _{1})&\chi _{2}(\mathbf {x} _{1})&\cdots &\chi _{N}(\mathbf {x} _{1})\\\chi _{1}(\mathbf {x} _{2})&\chi _{2}(\mathbf {x} _{2})&\cdots &\chi _{N}(\mathbf {x} _{2})\\\vdots &\vdots &\ddots &\vdots \\\chi _{1}(\mathbf {x} _{N})&\chi _{2}(\mathbf {x} _{N})&\cdots &\chi _{N}(\mathbf {x} _{N})\end{matrix}}\right|\equiv \left|{\begin{matrix}\chi _{1}&\chi _{2}&\cdots &\chi _{N}\\\end{matrix}}\right|,$

where in the final expression, a compact notation is introduced: the normalization constant and labels for the fermion coordinates are understood – only the wavefunctions are exhibited. The linear combination of Hartree products for the two-particle case can clearly be seen as identical with the Slater determinant for N = 2. It can be seen that the use of Slater determinants ensures an antisymmetrized function at the outset; symmetric functions are automatically rejected. In the same way, the use of Slater determinants ensures conformity to the w:Pauli principle. Indeed, the Slater determinant vanishes if the set {χ_i } is w:linearly dependent. In particular, this is the case when two (or more) spin orbitals are the same. In chemistry one expresses this fact by stating that no two electrons can occupy the same spin orbital. In general the Slater determinant is evaluated by the w:Laplace expansion. Mathematically, a Slater determinant is an antisymmetric tensor, also known as a w:wedge product.

A single Slater determinant is used as an approximation to the electronic wavefunction in Hartree–Fock theory. In more accurate theories (such as w:configuration interaction and w:MCSCF), a linear combination of Slater determinants is needed.

The word "detor" was proposed by S. F. Boys to describe the Slater determinant of the general type, but this term is rarely used.

Unlike w:fermions that are subject to the Pauli exclusion principle, two or more w:bosons can occupy the same quantum state of a system. Wavefunctions describing systems of identical w:bosons are symmetric under the exchange of particles and can be expanded in terms of w:permanents.

Fock matrix

In the w:Hartree–Fock method of w:quantum mechanics, the Fock matrix is a matrix approximating the single-electron w:energy operator of a given quantum system in a given set of basis vectors.

It is most often formed in w:computational chemistry when attempting to solve the w:Roothaan equations for an atomic or molecular system. The Fock matrix is actually an approximation to the true Hamiltonian operator of the quantum system. It includes the effects of electron-electron repulsion only in an average way. Importantly, because the Fock operator is a one-electron operator, it does not include the w:electron correlation energy.

The Fock matrix is defined by the Fock operator. For the restricted case which assumes w:closed-shell orbitals and single-determinantal wavefunctions, the Fock operator for the i-th electron is given by:

${\hat {F}}(i)={\hat {h}}(i)+\sum _{j=1}^{n/2}[2{\hat {J}}_{j}(i)-{\hat {K}}_{j}(i)]$

where:

${\hat {F}}(i)$ is the Fock operator for the i-th electron in the system,

${\hat {h}}(i)$ is the w:one-electron hamiltonian for the i-th electron,

$n$ is the number of electrons and ${\frac {n}{2}}$ is the number of occupied orbitals in the closed-shell system,

${\hat {J}}_{j}(i)$ is the w:Coulomb operator, defining the repulsive force between the j-th and i-th electrons in the system,

${\hat {K}}_{j}(i)$ is the w:exchange operator, defining the quantum effect produced by exchanging two electrons.

The Coulomb operator is multiplied by two since there are two electrons in each occupied orbital. The exchange operator is not multiplied by two since it has a non-zero result only for electrons which have the same spin as the i-th electron.

For systems with unpaired electrons there are many choices of Fock matrices.

Hartree-Fock (HF) or self-consistent field (SCF)

Density Functional Theory

Connection to quantum state symmetry

The Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to be antisymmetric. An antisymmetric two-particle state is represented as a sum of states in which one particle is in state $\scriptstyle |x\rangle$ and the other in state $\scriptstyle |y\rangle$:

$|\psi \rangle =\sum _{x,y}A(x,y)|x,y\rangle$

and antisymmetry under exchange means that A(x,y) = −A(y,x). This implies that A(x,x) = 0, which is Pauli exclusion. It is true in any basis, since unitary changes of basis keep antisymmetric matrices antisymmetric, although strictly speaking, the quantity A(x,y) is not a matrix but an antisymmetric rank-two w:tensor.

Conversely, if the diagonal quantities A(x,x) are zero in every basis, then the wavefunction component:

$A(x,y)=\langle \psi |x,y\rangle =\langle \psi |(|x\rangle \otimes |y\rangle )$

is necessarily antisymmetric. To prove it, consider the matrix element:

$\langle \psi |((|x\rangle +|y\rangle )\otimes (|x\rangle +|y\rangle ))\,$

This is zero, because the two particles have zero probability to both be in the superposition state $\scriptstyle |x\rangle +|y\rangle$. But this is equal to

$\langle \psi |x,x\rangle +\langle \psi |x,y\rangle +\langle \psi |y,x\rangle +\langle \psi |y,y\rangle \,$

The first and last terms on the right hand side are diagonal elements and are zero, and the whole sum is equal to zero. So the wavefunction matrix elements obey:

$\langle \psi |x,y\rangle +\langle \psi |y,x\rangle =0\,$.

or

$A(x,y)=-A(y,x)\,$

Pauli principle in advanced quantum theory

According to the w:spin-statistics theorem, particles with integer spin occupy symmetric quantum states, and particles with half-integer spin occupy antisymmetric states; furthermore, only integer or half-integer values of spin are allowed by the principles of quantum mechanics. In relativistic w:quantum field theory, the Pauli principle follows from applying a rotation operator in imaginary time to particles of half-integer spin. Since, nonrelativistically, particles can have any statistics and any spin, there is no way to prove a spin-statistics theorem in nonrelativistic quantum mechanics.

In one dimension, bosons, as well as fermions, can obey the exclusion principle. A one-dimensional Bose gas with delta function repulsive interactions of infinite strength is equivalent to a gas of free fermions. The reason for this is that, in one dimension, exchange of particles requires that they pass through each other; for infinitely strong repulsion this cannot happen. This model is described by a quantum w:nonlinear Schrödinger equation. In momentum space the exclusion principle is valid also for finite repulsion in a Bose gas with delta function interactions, as well as for interacting spins and w:Hubbard model in one dimension, and for other models solvable by w:Bethe ansatz. The ground state in models solvable by Bethe ansatz is a Fermi sphere.

Density Functional Theory

References

See also notes on editing this book about how to add references w:Nanotechnology/About#How_to_contribute.

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Modelling Nanosystems