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quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the variational method is one way of finding
approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
s to the lowest energy eigenstate or
ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
, and some excited states. This allows calculating approximate wavefunctions such as
molecular orbital In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of find ...
s. The basis for this method is the
variational principle In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those funct ...
. The method consists of choosing a "trial
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
" depending on one or more
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s, and finding the values of these parameters for which the
expectation value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of the energy is the lowest possible. The wavefunction obtained by fixing the parameters to such values is then an approximation to the ground state wavefunction, and the expectation value of the energy in that state is an upper bound to the ground state energy. The
Hartree–Fock method 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 ...
,
Density matrix renormalization group The density matrix renormalization group (DMRG) is a numerical variational technique devised to obtain the low-energy physics of quantum many-body systems with high accuracy. As a variational method, DMRG is an efficient algorithm that attempts ...
, and
Ritz method The Ritz method is a direct method to find an approximate solution for boundary value problems. The method is named after Walther Ritz, and is also commonly called the Rayleigh–Ritz method and the Ritz-Galerkin method. In quantum mechanics, a ...
apply the variational method.


Description

Suppose we are given a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
and a
Hermitian operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to i ...
over it called the Hamiltonian H . Ignoring complications about continuous spectra, we consider the discrete spectrum of H and a basis of eigenvectors \ (see spectral theorem for Hermitian operators for the mathematical background): \left\lang \psi_ , \psi_ \right\rang = \delta_, where \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
\delta_ = \begin 0 &\text i \neq j, \\ 1 &\text i=j, \end and the \ satisfy the eigenvalue equation H \left, \psi_\lambda\right\rangle = \lambda\left, \psi_\lambda \right\rangle. Once again ignoring complications involved with a continuous spectrum of H , suppose the spectrum of H is bounded from below and that its
greatest lower bound In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
is . The
expectation value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of H in a state , \psi\rangle is then \begin \left\langle\psi\ H \left, \psi\right\rangle & = \sum_ \left\langle\psi, \psi_\right\rangle \left\langle\psi_\H\left, \psi_\right\rangle \left\langle\psi_, \psi\right\rangle \\ & =\sum_\lambda \left, \left\langle\psi_\lambda , \psi\right\rangle\^2 \ge \sum_ E_0 \left, \left\langle\psi_\lambda , \psi\right\rangle\^2 = E_0 \langle \psi , \psi \rangle. \end If we were to vary over all possible states with norm 1 trying to minimize the expectation value of H , the lowest value would be E_0 and the corresponding state would be the ground state, as well as an eigenstate of H . Varying over the entire Hilbert space is usually too complicated for physical calculations, and a subspace of the entire Hilbert space is chosen, parametrized by some (real) differentiable parameters . The choice of the subspace is called the ansatz. Some choices of ansatzes lead to better approximations than others, therefore the choice of ansatz is important. Let's assume there is some overlap between the ansatz and the
ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
(otherwise, it's a bad ansatz). We wish to normalize the ansatz, so we have the constraints \left\langle \psi(\mathbf) , \psi(\mathbf) \right\rangle = 1 and we wish to minimize \varepsilon(\mathbf) = \left\langle \psi(\mathbf) \ H \left, \psi(\mathbf) \right\rangle. This, in general, is not an easy task, since we are looking for a
global minimum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
and finding the zeroes of the partial derivatives of over all is not sufficient. If is expressed as a linear combination of other functions ( being the coefficients), as in the
Ritz method The Ritz method is a direct method to find an approximate solution for boundary value problems. The method is named after Walther Ritz, and is also commonly called the Rayleigh–Ritz method and the Ritz-Galerkin method. In quantum mechanics, a ...
, there is only one minimum and the problem is straightforward. There are other, non-linear methods, however, such as the
Hartree–Fock method 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 ...
, that are also not characterized by a multitude of minima and are therefore comfortable in calculations. There is an additional complication in the calculations described. As tends toward in minimization calculations, there is no guarantee that the corresponding trial wavefunctions will tend to the actual wavefunction. This has been demonstrated by calculations using a modified harmonic oscillator as a model system, in which an exactly solvable system is approached using the variational method. A wavefunction different from the exact one is obtained by use of the method described above. Although usually limited to calculations of the ground state energy, this method can be applied in certain cases to calculations of excited states as well. If the ground state wavefunction is known, either by the method of variation or by direct calculation, a subset of the Hilbert space can be chosen which is orthogonal to the ground state wavefunction. \left, \psi \right\rangle = \left, \psi_\right\rangle - \left\langle\psi_ , \psi_\right\rangle \left, \psi_\right\rangle The resulting minimum is usually not as accurate as for the ground state, as any difference between the true ground state and \psi_ results in a lower excited energy. This defect is worsened with each higher excited state. In another formulation: E_\text \le \left\langle\phi\ H \left, \phi\right\rangle. This holds for any trial φ since, by definition, the ground state wavefunction has the lowest energy, and any trial wavefunction will have energy greater than or equal to it. Proof: can be expanded as a linear combination of the actual eigenfunctions of the Hamiltonian (which we assume to be normalized and orthogonal): \phi = \sum_n c_n \psi_n. Then, to find the expectation value of the Hamiltonian: \begin \left\langle H \right\rangle = \left\langle\phi\H\left, \phi\right\rangle = & \left\langle\sum_n c_n \psi_n \ H \left, \sum_m c_m\psi_m\right\rangle \\ = & \sum_n\sum_m \left\langle c_n^* \psi_\ E_m \left, c_m\psi_m\right\rangle \\ = & \sum_n\sum_m c_n^*c_m E_m\left\langle \psi_n , \psi_m \right\rangle \\ = & \sum_ , c_n, ^2 E_n. \end Now, the ground state energy is the lowest energy possible, i.e., E_ \ge E_. Therefore, if the guessed wave function is normalized: \left\langle\phi\ H \left, \phi\right\rangle \ge E_ \sum_n , c_n, ^2 = E_.


In general

For a hamiltonian ''H'' that describes the studied system and ''any'' normalizable function ''Ψ'' with arguments appropriate for the unknown wave function of the system, we define the
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...
\varepsilon\left Psi\right= \frac. The variational principle states that * \varepsilon \geq E_0, where E_0 is the lowest energy eigenstate (ground state) of the hamiltonian * \varepsilon = E_0 if and only if \Psi is exactly equal to the wave function of the ground state of the studied system. The variational principle formulated above is the basis of the variational method used in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
and
quantum chemistry Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
to find approximations to the
ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
. Another facet in variational principles in quantum mechanics is that since \Psi and \Psi^\dagger can be varied separately (a fact arising due to the complex nature of the wave function), the quantities can be varied in principle just one at a time.


Helium atom ground state

The helium atom consists of two
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
s with mass ''m'' and electric charge , around an essentially fixed
nucleus Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: * Atomic nucleus, the very dense central region of an atom *Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA Nucl ...
of mass and charge . The Hamiltonian for it, neglecting the
fine structure In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. It was first measured precisely for the hydrogen atom ...
, is: H = -\frac \left(\nabla_1^2 + \nabla_2^2\right) - \frac \left(\frac + \frac - \frac\right) where ''ħ'' is the
reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
, is the
vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
, (for ) is the distance of the -th electron from the nucleus, and is the distance between the two electrons. If the term , representing the repulsion between the two electrons, were excluded, the Hamiltonian would become the sum of two
hydrogen-like atom A hydrogen-like atom (or hydrogenic atom) is any atom or ion with a single valence electron. These atoms are isoelectronic with hydrogen. Examples of hydrogen-like atoms include, but are not limited to, hydrogen itself, all alkali metals such a ...
Hamiltonians with nuclear charge . The ground state energy would then be , where is the
Rydberg constant In spectroscopy, the Rydberg constant, symbol R_\infty for heavy atoms or R_\text for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectra of an atom. The constant first aro ...
, and its ground state wavefunction would be the product of two wavefunctions for the ground state of hydrogen-like atoms: \psi(\mathbf_1,\mathbf_2) = \frac e^. where is the
Bohr radius The Bohr radius (''a''0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an ...
and , helium's nuclear charge. The expectation value of the total Hamiltonian ''H'' (including the term ) in the state described by will be an upper bound for its ground state energy. is , so is . A tighter upper bound can be found by using a better trial wavefunction with 'tunable' parameters. Each electron can be thought to see the nuclear charge partially "shielded" by the other electron, so we can use a trial wavefunction equal with an "effective" nuclear charge : The expectation value of in this state is: \left\langle H \right\rangle = \left 2Z^2 + \frac Z\rightE_1 This is minimal for implying shielding reduces the effective charge to ~1.69. Substituting this value of into the expression for yields , within 2% of the experimental value, −78.975 eV. Even closer estimations of this energy have been found using more complicated trial wave functions with more parameters. This is done in physical chemistry via variational Monte Carlo.


References

{{Reflist, 2 Quantum chemistry Theoretical chemistry Computational chemistry Computational physics Approximations