In
chemical bond
A chemical bond is a lasting attraction between atoms or ions that enables the formation of molecules and crystals. The bond may result from the electrostatic force between oppositely charged ions as in ionic bonds, or through the sharing of ...
s, an orbital overlap is the concentration of
orbitals on adjacent atoms in the same regions of space. Orbital overlap can lead to bond formation.
Linus Pauling explained the importance of orbital overlap in the molecular
bond angle
Bond or bonds may refer to:
Common meanings
* Bond (finance), a type of debt security
* Bail bond, a commercial third-party guarantor of surety bonds in the United States
* Chemical bond, the attraction of atoms, ions or molecules to form chemical ...
s observed through experimentation; it is the basis for
orbital hybridization
In chemistry, orbital hybridisation (or hybridization) is the concept of mixing atomic orbitals to form new ''hybrid orbitals'' (with different energies, shapes, etc., than the component atomic orbitals) suitable for the pairing of electrons to ...
. As ''s'' orbitals are spherical (and have no directionality) and ''p'' orbitals are oriented 90° to each other, a theory was needed to explain why molecules such as
methane
Methane ( , ) is a chemical compound with the chemical formula (one carbon atom bonded to four hydrogen atoms). It is a group-14 hydride, the simplest alkane, and the main constituent of natural gas. The relative abundance of methane on Ea ...
(CH
4) had observed bond angles of 109.5°. Pauling proposed that s and p orbitals on the carbon atom can combine to form hybrids (sp
3 in the case of methane) which are directed toward the hydrogen atoms. The carbon hybrid orbitals have greater overlap with the hydrogen orbitals, and can therefore form stronger C–H bonds.
[Pauling, Linus. (1960). ''The Nature Of The Chemical Bond''. Cornell University Press.]
A quantitative measure of the overlap of two atomic orbitals Ψ
A and Ψ
B on atoms A and B is their overlap integral, defined as
:
where the integration extends over all space. The star on the first orbital wavefunction indicates the function's
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
, which in general may be
complex-valued.
Overlap matrix
The overlap matrix is a
square matrix, used in
quantum chemistry to describe the inter-relationship of a set of
basis vector
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
s of a
quantum system, such as an atomic orbital
basis set used in molecular electronic structure calculations. In particular, if the vectors are
orthogonal to one another, the overlap matrix will be diagonal. In addition, if the basis vectors form an
orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
set, the overlap matrix will be the
identity matrix. The overlap matrix is always ''n''×''n'', where ''n'' is the number of basis functions used. It is a kind of
Gramian matrix.
In general, each overlap matrix element is defined as an overlap integral:
:
where
:
is the ''j''-th basis
ket (
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
), and
:
is the ''j''-th
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 ...
, defined as :
.
In particular, if the set is normalized (though not necessarily orthogonal) then the diagonal elements will be identically 1 and the magnitude of the
off-diagonal element
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ� ...
s less than or equal to one with equality if and only if there is linear dependence in the basis set as per the
Cauchy–Schwarz inequality. Moreover, the matrix is always
positive definite; that is to say, the eigenvalues are all strictly positive.
See also
*
Roothaan equations
*
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 ...
*
Pi bond
In chemistry, pi bonds (π bonds) are covalent chemical bonds, in each of which two lobes of an orbital on one atom overlap with two lobes of an orbital on another atom, and in which this overlap occurs laterally. Each of these atomic orbitals ...
*
Sigma bond
In chemistry, sigma bonds (σ bonds) are the strongest type of covalent chemical bond. They are formed by head-on overlapping between atomic orbitals. Sigma bonding is most simply defined for diatomic molecules using the language and tools of s ...
References
''Quantum Chemistry: Fifth Edition'', Ira N. Levine, 2000
Quantum chemistry
Matrices
Chemical bonding
Molecular geometry
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