Walsh diagram
Encyclopedia
Walsh diagrams, often called angular coordinate diagrams or correlation diagrams, are representations of calculated orbital energies of a molecule
versus a distortion coordinate, used for making quick predictions about the geometries of small molecules. By plotting the change in molecular orbital
levels of a molecule as a function of geometrical change, Walsh diagrams explain why molecules are more stable in certain spatial configurations (i.e. why water adopts a bent conformation).
A major application of Walsh diagrams is to explain the regularity in structure observed for related molecules having identical numbers of valence electrons (i.e. why H2O and H2S look similar), and to account for how molecules alter their geometries as their number of electrons or spin state
changes. Additionally, Walsh diagrams can be used to predict distortions of molecular geometry from knowledge of how the LUMO
(Lowest Unoccupied Molecular Orbital) affects the HOMO
(Highest Occupied Molecular Orbital) when the molecule experiences geometrical perturbation.
Walsh’s rule for predicting shapes of molecules states that a molecule will adopt a structure that best provides the most stability for its HOMO. If a particular structural change does not perturb the HOMO, the closest occupied molecular orbital governs the preference for geometrical orientation.
, a British chemistry professor at the University of Dundee
, in a series of papers in one issue of the Journal of the Chemical Society. Here, he aimed to rationalize the shapes adopted by polyatomic molecules in the ground state
as well as in excited state
s, by applying theoretical contributions made by Mulliken
. Specifically, Walsh calculated and explained the effect of changes in the shape of a molecule on the energy of molecular orbitals. Walsh diagrams are an illustration of such dependency, and his conclusions are what are referred to as the “rules of Walsh.”
In his publications, Walsh showed through multiple examples that the geometry adopted by a molecule in its ground state primarily depends on the number of its valence electrons. He himself acknowledged that this general concept was not novel, but explained that the new data available to him allowed the previous generalizations to be expanded upon and honed. He also noted that Mulliken had previously attempted to construct a correlation diagram for the possible orbitals of a polyatomic molecule in two different nuclear configurations, and had even tried to use this diagram to explain shapes and spectra
of molecules in their ground and excited states. However, Mulliken was unable to explain the reasons for the rises and falls of certain curves with increases in angle, thus Walsh claimed “his diagram was either empirical or based upon unpublished computations.”
. What Walsh was actually describing by this term is unclear; some believe he was in fact referring to ionization potentials
, however this remains a topic of debate. At any rate, the general concept he put forth was that the total energy of a molecule is equal to the sum of all of the “orbital binding energies” in that molecule. Hence, from knowledge of the stabilization or destabilization of each of the orbitals by an alteration of the molecular bond angle, the equilibrium bond angle for a particular state of the molecule can be predicted. Orbitals which interact to stabilize one configuration (ex. Linear) may or may not overlap in another configuration (ex. Bent), thus one geometry will be calculably more stable than the other.
Typically, core orbitals (1s for B, C, N, O, F, and Ne) are excluded from Walsh diagrams because they are so low in energy that they do not experience a significant change by variations in bond angle. Only valence orbitals are considered. However, one should keep in mind that some of the valence orbitals are often unoccupied.
s of a molecule for a specific arrangement of the atoms in that molecule. In conducting these calculations, bond lengths remain constant (at equilibrium
values) and only the bond angle should be altered from its equilibrium value. The single-point computation for each geometry can then be plotted versus bond angle to produce the representative Walsh diagram.
orbital energy curves for the canonical molecular orbitals while changing the bond angle from 90° to 180°. As the bond angle is distorted, the energy for each of the orbitals can be followed along the lines, allowing a quick approximation of molecular energy as a function of conformation. It is still unclear whether or not the Walsh ordinate considers nuclear repulsion, and this remains a topic of debate.
This same concept can be applied to other species including non-hydride AB2 and BAC molecules, HAB and HAAH molecules, tetraatomic hydride molecules (AH3), tetraatomic nonhydride molecules (AB), H2AB molecules, acetaldehyde
, pentaatomic molecules (CH3I), hexaatomic molecules (ethylene
), and benzene
.
can also be used as a tool to predict reactivity. By generating a Walsh Diagram and then determining the HOMO/LUMO of that molecule, it can be determined how the molecule is likely to react. In the following example, the Lewis acid
ity of AH3 molecules such as BH3 and CH3+ is predicted.
Six electron AH3 molecules should have a planar conformation. It can be seen that the HOMO, 1e’, of planar AH3 is destabilized upon bending of the A-H bonds to form a pyramid shape, due to disruption of bonding. The LUMO, which is concentrated on one atomic center, is a good electron acceptor and explains the Lewis acid character of BH3 and CH3+.
Walsh correlation diagrams can also be used to predict relative molecular orbital
energy levels. The distortion of the hydrogen atoms from the planar CH3+ to the tetrahedral CH3-Nu causes a stabilization of the C-Nu bonding orbital, σ.
s and Orgel diagrams.
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
versus a distortion coordinate, used for making quick predictions about the geometries of small molecules. By plotting the change in molecular orbital
Molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the 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 finding an electron in any specific region. The term "orbital" was first...
levels of a molecule as a function of geometrical change, Walsh diagrams explain why molecules are more stable in certain spatial configurations (i.e. why water adopts a bent conformation).
A major application of Walsh diagrams is to explain the regularity in structure observed for related molecules having identical numbers of valence electrons (i.e. why H2O and H2S look similar), and to account for how molecules alter their geometries as their number of electrons or spin state
Spin quantum number
In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given particle...
changes. Additionally, Walsh diagrams can be used to predict distortions of molecular geometry from knowledge of how the LUMO
Lumo
Lumo is a 2007 documentary film about twenty-year-old Lumo Sinai, a woman who fell victim to "Africa's First World War." While returning home one day, Lumo and another woman were gang-raped by a group of soldiers fighting for control of the Democratic Republic of the Congo during the 1994 Rwandan...
(Lowest Unoccupied Molecular Orbital) affects the HOMO
Homo
Homo may refer to:*the Greek prefix ὅμο-, meaning "the same"*the Latin for man, human being*Homo, the taxonomical genus including modern humans...
(Highest Occupied Molecular Orbital) when the molecule experiences geometrical perturbation.
Walsh’s rule for predicting shapes of molecules states that a molecule will adopt a structure that best provides the most stability for its HOMO. If a particular structural change does not perturb the HOMO, the closest occupied molecular orbital governs the preference for geometrical orientation.
History
Walsh diagrams were first introduced by A.D. WalshA.D. Walsh
Arthur Donald Walsh FRS was a British chemist, professor of chemistry at the University of Dundee. He was elected FRS in 1964. He was educated at Loughborough Grammar School.- Biography :...
, a British chemistry professor at the University of Dundee
University of Dundee
The University of Dundee is a university based in the city and Royal burgh of Dundee on eastern coast of the central Lowlands of Scotland and with a small number of institutions elsewhere....
, in a series of papers in one issue of the Journal of the Chemical Society. Here, he aimed to rationalize the shapes adopted by polyatomic molecules in the ground state
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; 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...
as well as in excited state
Excited state
Excitation is an elevation in energy level above an arbitrary baseline energy state. In physics there is a specific technical definition for energy level which is often associated with an atom being excited to an excited state....
s, by applying theoretical contributions made by Mulliken
Robert S. Mulliken
Robert Sanderson Mulliken was an American physicist and chemist, primarily responsible for the early development of molecular orbital theory, i.e. the elaboration of the molecular orbital method of computing the structure of molecules. Dr. Mulliken received the Nobel Prize for chemistry in 1966...
. Specifically, Walsh calculated and explained the effect of changes in the shape of a molecule on the energy of molecular orbitals. Walsh diagrams are an illustration of such dependency, and his conclusions are what are referred to as the “rules of Walsh.”
In his publications, Walsh showed through multiple examples that the geometry adopted by a molecule in its ground state primarily depends on the number of its valence electrons. He himself acknowledged that this general concept was not novel, but explained that the new data available to him allowed the previous generalizations to be expanded upon and honed. He also noted that Mulliken had previously attempted to construct a correlation diagram for the possible orbitals of a polyatomic molecule in two different nuclear configurations, and had even tried to use this diagram to explain shapes and spectra
Spectrum
A spectrum is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by...
of molecules in their ground and excited states. However, Mulliken was unable to explain the reasons for the rises and falls of certain curves with increases in angle, thus Walsh claimed “his diagram was either empirical or based upon unpublished computations.”
Overview
Walsh originally constructed his diagrams by plotting what he described as “orbital binding energies” versus bond anglesMolecular geometry
Molecular geometry or molecular structure is the three-dimensional arrangement of the atoms that constitute a molecule. It determines several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism, and biological activity.- Molecular geometry determination...
. What Walsh was actually describing by this term is unclear; some believe he was in fact referring to ionization potentials
Ionization energy
The ionization energy of a chemical species, i.e. an atom or molecule, is the energy required to remove an electron from the species to a practically infinite distance. Large atoms or molecules have a low ionization energy, while small molecules tend to have higher ionization energies.The property...
, however this remains a topic of debate. At any rate, the general concept he put forth was that the total energy of a molecule is equal to the sum of all of the “orbital binding energies” in that molecule. Hence, from knowledge of the stabilization or destabilization of each of the orbitals by an alteration of the molecular bond angle, the equilibrium bond angle for a particular state of the molecule can be predicted. Orbitals which interact to stabilize one configuration (ex. Linear) may or may not overlap in another configuration (ex. Bent), thus one geometry will be calculably more stable than the other.
Typically, core orbitals (1s for B, C, N, O, F, and Ne) are excluded from Walsh diagrams because they are so low in energy that they do not experience a significant change by variations in bond angle. Only valence orbitals are considered. However, one should keep in mind that some of the valence orbitals are often unoccupied.
Generating Walsh Diagrams
In preparing a Walsh diagram, the geometry of a molecule must first be optimized using the Hartree-Fock (HF) method for approximating the ground-state wave function and ground-state energy of a quantum many-body system. Next, single-point energies are performed for a series of geometries displaced from the above-determined equilibrium geometry. Single-point energies (SPEs) are calculations of potential energy surfacePotential energy surface
A potential energy surface is generally used within the adiabatic or Born–Oppenheimer approximation in quantum mechanics and statistical mechanics to model chemical reactions and interactions in simple chemical and physical systems...
s of a molecule for a specific arrangement of the atoms in that molecule. In conducting these calculations, bond lengths remain constant (at equilibrium
Equilibrium
Equilibrium is the condition of a system in which competing influences are balanced. The word may refer to:-Biology:* Equilibrioception, the sense of a balance present in human beings and other animals...
values) and only the bond angle should be altered from its equilibrium value. The single-point computation for each geometry can then be plotted versus bond angle to produce the representative Walsh diagram.
Structure of a Walsh Diagram
AH2 Molecules
For the simplest AH2 molecular system, Walsh produced the first angular correlation diagram by plotting the ab initioAb initio
ab initio is a Latin term used in English, meaning from the beginning.ab initio may also refer to:* Ab Initio , a leading ETL Tool Software Company in the field of Data Warehousing.* ab initio quantum chemistry methods...
orbital energy curves for the canonical molecular orbitals while changing the bond angle from 90° to 180°. As the bond angle is distorted, the energy for each of the orbitals can be followed along the lines, allowing a quick approximation of molecular energy as a function of conformation. It is still unclear whether or not the Walsh ordinate considers nuclear repulsion, and this remains a topic of debate.
This same concept can be applied to other species including non-hydride AB2 and BAC molecules, HAB and HAAH molecules, tetraatomic hydride molecules (AH3), tetraatomic nonhydride molecules (AB), H2AB molecules, acetaldehyde
Acetaldehyde
Acetaldehyde is an organic chemical compound with the formula CH3CHO or MeCHO. It is one of the most important aldehydes, occurring widely in nature and being produced on a large scale industrially. Acetaldehyde occurs naturally in coffee, bread, and ripe fruit, and is produced by plants as part...
, pentaatomic molecules (CH3I), hexaatomic molecules (ethylene
Ethylene
Ethylene is a gaseous organic compound with the formula . It is the simplest alkene . Because it contains a carbon-carbon double bond, ethylene is classified as an unsaturated hydrocarbon. Ethylene is widely used in industry and is also a plant hormone...
), and benzene
Benzene
Benzene is an organic chemical compound. It is composed of 6 carbon atoms in a ring, with 1 hydrogen atom attached to each carbon atom, with the molecular formula C6H6....
.
Reactivity
Walsh diagrams in conjunction with molecular orbital theoryMolecular orbital theory
In chemistry, molecular orbital theory is a method for determining molecular structure in which electrons are not assigned to individual bonds between atoms, but are treated as moving under the influence of the nuclei in the whole molecule...
can also be used as a tool to predict reactivity. By generating a Walsh Diagram and then determining the HOMO/LUMO of that molecule, it can be determined how the molecule is likely to react. In the following example, the Lewis acid
Lewis acid
]The term Lewis acid refers to a definition of acid published by Gilbert N. Lewis in 1923, specifically: An acid substance is one which can employ a lone pair from another molecule in completing the stable group of one of its own atoms. Thus, H+ is a Lewis acid, since it can accept a lone pair,...
ity of AH3 molecules such as BH3 and CH3+ is predicted.
Six electron AH3 molecules should have a planar conformation. It can be seen that the HOMO, 1e’, of planar AH3 is destabilized upon bending of the A-H bonds to form a pyramid shape, due to disruption of bonding. The LUMO, which is concentrated on one atomic center, is a good electron acceptor and explains the Lewis acid character of BH3 and CH3+.
Walsh correlation diagrams can also be used to predict relative molecular orbital
Molecular orbital theory
In chemistry, molecular orbital theory is a method for determining molecular structure in which electrons are not assigned to individual bonds between atoms, but are treated as moving under the influence of the nuclei in the whole molecule...
energy levels. The distortion of the hydrogen atoms from the planar CH3+ to the tetrahedral CH3-Nu causes a stabilization of the C-Nu bonding orbital, σ.
Other correlation diagrams
Other correlation diagrams are Tanabe-Sugano diagramTanabe-Sugano diagram
Tanabe-Sugano diagrams are used in coordination chemistry to predict absorptions in the UV and visible electromagnetic spectrum of coordination compounds. The results from a Tanabe-Sugano diagram analysis of a metal complex can also be compared to experimental spectroscopic data...
s and Orgel diagrams.
See also
- Ab initio quantum chemistry methodsAb initio quantum chemistry methodsAb initio quantum chemistry methods are computational chemistry methods based on quantum chemistry. The term ab initiowas first used in quantum chemistry by Robert Parr and coworkers, including David Craig in a semiempirical study on the excited states of benzene.The background is described by Parr...
- Molecular Orbital TheoryMolecular orbital theoryIn chemistry, molecular orbital theory is a method for determining molecular structure in which electrons are not assigned to individual bonds between atoms, but are treated as moving under the influence of the nuclei in the whole molecule...
- Mulliken symbols