Configuration state function

Last updated

In quantum chemistry, a configuration state function (CSF), is a symmetry-adapted linear combination of Slater determinants. A CSF must not be confused with a configuration. In general, one configuration gives rise to several CSFs; all have the same total quantum numbers for spin and spatial parts but differ in their intermediate couplings.

Contents

Definition

A configuration state function (CSF), is a symmetry-adapted linear combination of Slater determinants. It is constructed to have the same quantum numbers as the wavefunction, , of the system being studied. In the method of configuration interaction, the wavefunction [1] can be expressed as a linear combination of CSFs, that is in the form

where denotes the set of CSFs. The coefficients, , are found by using the expansion of to compute a Hamiltonian matrix. When this is diagonalized, the eigenvectors are chosen as the expansion coefficients. CSFs rather than just Slater determinants can also be used as a basis in multi-configurational self-consistent field computations.

In atomic structure, a CSF is an eigenstate of

In linear molecules, does not commute with the Hamiltonian for the system and therefore CSFs are not eigenstates of . However, the z-projection of angular momentum is still a good quantum number and CSFs are constructed to be eigenstates of and . In non-linear (which implies polyatomic) molecules, neither nor commutes with the Hamiltonian. The CSFs are constructed to have the spatial transformation properties of one of the irreducible representations of the point group to which the nuclear framework belongs. This is because the Hamiltonian operator transforms in the same way. [2] and are still valid quantum numbers and CSFs are built to be eigenfunctions of these operators.

From configurations to configuration state functions

CSFs are derived from configurations. A configuration is just an assignment of electrons to orbitals. For example, and are example of two configurations, one from atomic structure and one from molecular structure.

From any given configuration we can, in general, create several CSFs. CSFs are therefore sometimes also called N-particle symmetry adapted basis functions. For a configuration the number of electrons is fixed; let's call this . When we are creating CSFs from a configuration we have to work with the spin-orbitals associated with the configuration.

For example, given the orbital in an atom we know that there are two spin-orbitals associated with this,

where

are the one electron spin-eigenfunctions for spin-up and spin-down respectively. Similarly, for the orbital in a linear molecule ( point group) we have four spin orbitals:

.

This is because the designation corresponds to z-projection of angular momentum of both and .

We can think of the set of spin orbitals as a set of boxes each of size one; let's call this boxes. We distribute the electrons among the boxes in all possible ways. Each assignment corresponds to one Slater determinant, . There can be great number of these, particularly when . Another way to look at this is to say we have entities and we wish to select of them, known as a combination. We need to find all possible combinations. Order of the selection is not significant because we are working with determinants and can interchange rows as required.

If we then specify the overall coupling that we wish to achieve for the configuration, we can now select only those Slater determinants that have the required quantum numbers. In order to achieve the required total spin angular momentum (and in the case of atoms the total orbital angular momentum as well), each Slater determinant has to be premultiplied by a coupling coefficient , derived ultimately from Clebsch–Gordan coefficients. Thus the CSF is a linear combination

.

The Lowdin projection operator formalism [3] may be used to find the coefficients. For any given set of determinants it may be possible to find several different sets of coefficients. [4] Each set corresponds to one CSF. In fact this simply reflects the different internal couplings of total spin and spatial angular momentum.

A genealogical algorithm for CSF construction

At the most fundamental level, a configuration state function can be constructed from a set of orbitals and a number of electrons using the following genealogical algorithm:

  1. distribute the electrons over the set of orbitals giving a configuration
  2. for each orbital the possible quantum number couplings (and therefore wavefunctions for the individual orbitals) are known from basic quantum mechanics; for each orbital choose one of the permitted couplings but leave the z-component of the total spin, undefined.
  3. check that the spatial coupling of all orbitals matches that required for the system wavefunction. For a molecule exhibiting or this is achieved by a simple linear summation of the coupled value for each orbital; for molecules whose nuclear framework transforms according to symmetry, or one of its sub-groups, the group product table has to be used to find the product of the irreducible representation of all orbitals.
  4. couple the total spins of the orbitals from left to right; this means we have to choose a fixed for each orbital.
  5. test the final total spin and its z-projection against the values required for the system wavefunction

The above steps will need to be repeated many times to elucidate the total set of CSFs that can be derived from the electrons and orbitals.

Single orbital configurations and wavefunctions

Basic quantum mechanics defines the possible single orbital wavefunctions. In a software implementation, these can be provided either as a table or through a set of logic statements. Alternatively group theory may be used to compute them. [5] Electrons in a single orbital are called equivalent electrons. [6] They obey the same coupling rules as other electrons but the Pauli exclusion principle makes certain couplings impossible. The Pauli exclusion principle requires that no two electrons in a system can have all their quantum numbers equal. For equivalent electrons, by definition the principal quantum number is identical. In atoms the angular momentum is also identical. So, for equivalent electrons the z components of spin and spatial parts, taken together, must differ.

The following table shows the possible couplings for a orbital with one or two electrons.

Orbital ConfigurationTerm symbol projection

The situation for orbitals in Abelian point groups mirrors the above table. The next table shows the fifteen possible couplings for a orbital. The orbitals also each generate fifteen possible couplings, all of which can be easily inferred from this table.

Orbital ConfigurationTerm symbolLambda coupling projection

Similar tables can be constructed for atomic systems, which transform according to the point group of the sphere, that is for s, p, d, f orbitals. The number of term symbols and therefore possible couplings is significantly larger in the atomic case.

Computer Software for CSF generation

Computer programs are readily available to generate CSFs for atoms [7] for molecules [8] and for electron and positron scattering by molecules. [9] A popular computational method for CSF construction is the Graphical Unitary Group Approach.

Related Research Articles

<span class="mw-page-title-main">Atomic orbital</span> Function describing an electron in an atom

In quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function describes the electron's charge distribution around the atom's nucleus, and can be used to calculate the probability of finding an electron in a specific region around the nucleus.

<span class="mw-page-title-main">Hydrogen atom</span> Atom of the element hydrogen

A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen constitutes about 75% of the baryonic mass of the universe.

<span class="mw-page-title-main">Energy level</span> Different states of quantum systems

A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The term is commonly used for the energy levels of the electrons in atoms, ions, or molecules, which are bound by the electric field of the nucleus, but can also refer to energy levels of nuclei or vibrational or rotational energy levels in molecules. The energy spectrum of a system with such discrete energy levels is said to be quantized.

<span class="mw-page-title-main">Hyperfine structure</span> Small shifts and splittings in the energy levels of atoms, molecules and ions

In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate electronic energy levels and the resulting splittings in those electronic energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the nucleus and electron clouds.

<span class="mw-page-title-main">Azimuthal quantum number</span> Quantum number denoting orbital angular momentum

In quantum mechanics, the azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes aspects of the angular shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe the unique quantum state of an electron.

<span class="mw-page-title-main">Magnetic moment</span> Magnetic strength and orientation of an object that produces a magnetic field

In electromagnetism, the magnetic moment or magnetic dipole moment is the combination of strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude of torque the object experiences in a given magnetic field. When the same magnetic field is applied, objects with larger magnetic moments experience larger torques. The strength of this torque depends not only on the magnitude of the magnetic moment but also on its orientation relative to the direction of the magnetic field. Its direction points from the south pole to north pole of the magnet.

In atomic physics, a magnetic quantum number is a quantum number used to distinguish quantum states of an electron or other particle according to its angular momentum along a given axis in space. The orbital magnetic quantum number distinguishes the orbitals available within a given subshell of an atom. It specifies the component of the orbital angular momentum that lies along a given axis, conventionally called the z-axis, so it describes the orientation of the orbital in space. The spin magnetic quantum numberms specifies the z-axis component of the spin angular momentum for a particle having spin quantum number s. For an electron, s is 12, and ms is either +12 or −12, often called "spin-up" and "spin-down", or α and β. The term magnetic in the name refers to the magnetic dipole moment associated with each type of angular momentum, so states having different magnetic quantum numbers shift in energy in a magnetic field according to the Zeeman effect.

<span class="mw-page-title-main">Fine structure</span> Details in the emission spectrum of an atom

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 by Albert A. Michelson and Edward W. Morley in 1887, laying the basis for the theoretical treatment by Arnold Sommerfeld, introducing the fine-structure constant.

In physics and chemistry, a selection rule, or transition rule, formally constrains the possible transitions of a system from one quantum state to another. Selection rules have been derived for electromagnetic transitions in molecules, in atoms, in atomic nuclei, and so on. The selection rules may differ according to the technique used to observe the transition. The selection rule also plays a role in chemical reactions, where some are formally spin-forbidden reactions, that is, reactions where the spin state changes at least once from reactants to products.

In physics, the C parity or charge parity is a multiplicative quantum number of some particles that describes their behavior under the symmetry operation of charge conjugation.

In molecular physics, the molecular term symbol is a shorthand expression of the group representation and angular momenta that characterize the state of a molecule, i.e. its electronic quantum state which is an eigenstate of the electronic molecular Hamiltonian. It is the equivalent of the term symbol for the atomic case. However, the following presentation is restricted to the case of homonuclear diatomic molecules, or other symmetric molecules with an inversion centre. For heteronuclear diatomic molecules, the u/g symbol does not correspond to any exact symmetry of the electronic molecular Hamiltonian. In the case of less symmetric molecules the molecular term symbol contains the symbol of the group representation to which the molecular electronic state belongs.

<span class="mw-page-title-main">Orbital motion (quantum)</span> Quantum mechanical property

Quantum orbital motion involves the quantum mechanical motion of rigid particles about some other mass, or about themselves. In classical mechanics, an object's orbital motion is characterized by its orbital angular momentum and spin angular momentum, which is the object's angular momentum about its own center of mass. In quantum mechanics there are analogous orbital and spin angular momenta which describe the orbital motion of a particle, represented as quantum mechanical operators instead of vectors.

Multi-configurational self-consistent field (MCSCF) is a method in quantum chemistry used to generate qualitatively correct reference states of molecules in cases where Hartree–Fock and density functional theory are not adequate. It uses a linear combination of configuration state functions (CSF), or configuration determinants, to approximate the exact electronic wavefunction of an atom or molecule. In an MCSCF calculation, the set of coefficients of both the CSFs or determinants and the basis functions in the molecular orbitals are varied to obtain the total electronic wavefunction with the lowest possible energy. This method can be considered a combination between configuration interaction and Hartree–Fock.

Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

In atomic, molecular, and optical physics and quantum chemistry, the molecular Hamiltonian is the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule. This operator and the associated Schrödinger equation play a central role in computational chemistry and physics for computing properties of molecules and aggregates of molecules, such as thermal conductivity, specific heat, electrical conductivity, optical, and magnetic properties, and reactivity.

In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-1/2 particles, which takes into account the interaction of the particle's spin with an external electromagnetic field. It is the non-relativistic limit of the Dirac equation and can be used where particles are moving at speeds much less than the speed of light, so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli in 1927. In its linearized form it is known as Lévy-Leblond equation.

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 as Rb and Cs, singly ionized alkaline earth metals such as Ca+ and Sr+ and other ions such as He+, Li2+, and Be3+ and isotopes of any of the above. A hydrogen-like atom includes a positively charged core consisting of the atomic nucleus and any core electrons as well as a single valence electron. Because helium is common in the universe, the spectroscopy of singly ionized helium is important in EUV astronomy, for example, of DO white dwarf stars.

In 1927, a year after the publication of the Schrödinger equation, Hartree formulated what are now known as the Hartree equations for atoms, using the concept of self-consistency that Lindsay had introduced in his study of many electron systems in the context of Bohr theory. Hartree assumed that the nucleus together with the electrons formed a spherically symmetric field. The charge distribution of each electron was the solution of the Schrödinger equation for an electron in a potential , derived from the field. Self-consistency required that the final field, computed from the solutions, was self-consistent with the initial field, and he thus called his method the self-consistent field method.

<span class="mw-page-title-main">Helium atom</span> Atom of helium

A helium atom is an atom of the chemical element helium. Helium is composed of two electrons bound by the electromagnetic force to a nucleus containing two protons along with two neutrons, depending on the isotope, held together by the strong force. Unlike for hydrogen, a closed-form solution to the Schrödinger equation for the helium atom has not been found. However, various approximations, such as the Hartree–Fock method, can be used to estimate the ground state energy and wavefunction of the atom. Historically, the first such helium spectrum calculation was done by Albrecht Unsöld in 1927. Its success was considered to be one of the earliest signs of validity of Schrödinger's wave mechanics.

In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator.

References

  1. Engel, T. (2006). Quantum Chemistry and Spectroscopy. Pearson PLC. ISBN   0-8053-3842-X.
  2. Pilar, F. L. (1990). Elementary Quantum Chemistry (2nd ed.). Dover Publications. ISBN   0-486-41464-7.
  3. Crossley, R. J. S. (1977). "On Löwdin's projection operators for angular momentum. I". International Journal of Quantum Chemistry . 11 (6): 917–929. doi:10.1002/qua.560110605.
  4. Nesbet, R. K. (2003). "Section 4.4". In Huo, W. M .; Gianturco, F. A. (eds.). Variational principles and methods in theoretical physics and chemistry. Cambridge University Press. p. 49. ISBN   0-521-80391-8.
  5. Karayianis, N. (1965). "Atomic Terms for Equivalent Electrons". J. Math. Phys. 6 (8): 1204–1209. Bibcode:1965JMP.....6.1204K. doi:10.1063/1.1704761.
  6. Wise, J.H. (1976). "Spectroscopic terms for equivalent electrons". J. Chem. Educ. 53 (8): 496. Bibcode:1976JChEd..53..496W. doi:10.1021/ed053p496.2.
  7. Sturesson, L.; Fischer, C. F. (1993). "LSGEN - a program to generate configuration-state lists of LS-coupled basis functions". Computer Physics Communications . 74 (3): 432–440. Bibcode:1993CoPhC..74..432S. doi:10.1016/0010-4655(93)90024-7.
  8. McLean, A. D.; et al. (1991). "ALCHEMY II, A Research Tool for Molecular Electronic Structure and Interactions". In Clementi, E. (ed.). Modern Techniques in Computational Chemistry (MOTECC-91). ESCOM Science Publishers. ISBN   90-72199-10-3.
  9. Morgan, L. A.; Tennyson, J.; Gillan, C. J. (1998). "The UK molecular R-matrix codes". Computer Physics Communications . 114 (1–3): 120–128. Bibcode:1998CoPhC.114..120M. doi:10.1016/S0010-4655(98)00056-3.