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As if the theory of the chemical bond isn't difficult enough, it is also plagued by different people using different notations and different terminology for more or less the same things. So if you're confused, maybe the following may be helpful.
Variables.
Equations in quantum chemistry are often long and complicated. Therefore quite a number of notations have been devised to shorten them. Often these notations sacrifice clarity for brevity. This is the case with the way variables of wave functions and integration variables are written. The position of an electron is written in full as (x,y,z) or more compact as r. If there are more electrons a subscript is added: rn stands for the position of electron n. The variable corresponding to the spin of an electron is generally written as s (sn). This should not be confused with a wave function representing only the spin. These wave functions are a and b, or with the spin variable a(s) and b(s). The position and spin of an electron are often combined to t (tn).(This symbol is also used for all coordinates and spins of a system of more than one electron.) If there is more than one electron t is often left out, and only a number is used indicating the electron. So we have
| Y(1,2,...,N) = Y(t1,t2,...,tN) |
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for wave functions and
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ó
õ |
d1 d2...dN ... = |
ó
õ |
dt1 dt2...dtN ... |
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for integrals. This same notation is also used, however, for (spatial) orbitals that do not include the spin of the electron. In that case we have we have
| F(1,2,...,N) = F(r1,r2,...,rN) |
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for wave functions and
|
ó
õ |
d1 d2...dN ... = |
ó
õ |
dr1 dr2...drN ... |
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for integrals. The variables of wave functions are often left out altogether.
The Dirac bracket notation.
The Dirac bracket notation, or just bracket notation, is a short hand notation that is used in this course only for integrals. The name derives from the angular brackets á and ñ that are used and the person, P.A.M. Dirac, who introduced the notation. A matrix element in the bracket notation is defined as
| áY|A|Fñ = |
ó
õ |
dt Y(t)*AF(t). |
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In this definition Y and F are wave functions and A is an operator. The integration variable t stands for all coordinates and spins of the particles that form the system described by the wave functions. The reason why this is called a matrix element is that such integrals often define elements of matrices that are encountered in quantum mechanics. Integrals with just wave functions and no operator are written as
| áY|Fñ = |
ó
õ |
dt Y(t)*F(t). |
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Such a quantity is called an inner product, because it has the properties of an inner product. The bracket notation leaves out the integration variables. These variables have to be inferred from the wave functions.
The part on the left á...| is called a "bra,'' and the part on the right |...ñ a "ket.'' (Together they form a bra(c)ket.) In a formal treatment of quantum mechanics bras and kets are not only used in combination as matrix elements and inner products, but also separately. As such they become very powerful tools to make otherwise very complicated derivations almost trivial.
The independent-particle model, MO theory, the Hartree-Fock approximation, and SCF.
In quantum chemistry the independent-particle model, MO theory, the Hartree-Fock approximation, and SCF all refer more or less to the same thing. The term "independent-particle model'' is used to indicate a model in which a particle feels only the field generated by the other particles averaged over the positions (and possibly spins) of these other particles. The standard wave function for such a model is a product Y(1,2,...,N) = y1(1)y2(2)...yN(N), because such a function has no correlation in the positions of the particles. The absolute square of the wave function is a product of the absolute squares of the single-particle functions (orbitals if the particles are electrons): |Y(1,2,...,N)|2 = |y1(1)|2|y2(2)|2 ... |yN(N)|2. This means that the particles move independently. Hence the name of the model. Because of the Pauli principle for electrons the independent-particle model for electrons uses a Slater determinant. Such a Slater determinant has also no correlation in position between electrons with different spins, but there is correlation between electrons with the same spin, because such electrons cannot be at the same position at the same time. This can be seen by substituting the same position for two electrons with the same spin in a Slater determinant. Two columns will become the same, which means that the Slater determinant is zero. The functions in a Slater determinant are called orbitals, and, if we're dealing with a molecule, molecular orbitals or MO's. Hence the independent-particle model is also called molecular orbital theory or MO theory.
Yet another term is "Hartree-Fock approximation'' honoring Hartree and Fock who contributed much to the theory of electron systems in the 1920's and 1930's. Strictly speaking "Hartree-Fock approximation'' is a more restrictive term than "MO theory.'' In the Hartree-Fock approximation the use of a Slater determinant for the wave function and usually the LCAO approximation for the orbitals are the only approximations. All calculations are done exactly. There are various forms of MO theory, however, in which some integrals are approximated to speed up the computation so that larger molecules can be handled. The self-consistent field (SCF) approximation is another name for the Hartree-Fock approximation. The name refers to the procedure to solve the Fock equation. (This is outside the scope of this course.) As this procedure is used in other methods as well, there are more SCF approximations.
The Fock equation, the Roothaan equation, and the secular equation.
The Fock, Roothaan, and secular equation are essentially the same equation. The Fock equation is given by
If we use the LCAO approximation then this equation becomes a matrix eigenvalue equation, which is called the Roothaan equation or the secular equation. The latter name is used more often, but the former is to be preferred. The term secular equation is used in mathematics as an alternative to eigenvalue equation, and has no physical or chemical meaning. In fact, there are many secular equations in quantum chemistry.
Different types of orbitals.
A molecular orbital (MO) is any wave function for one electron in a molecule. Some molecular orbitals are special. A canonical MO is a solution of a Fock equation. These are very important MO's. In fact the phrase "the MO's of a molecule'' is often used synonymously to "the canonical MO's of a molecule.'' But canonical MO's are certainly not the only MO's used to describe bonding in molecules. Hybrids are MO's that are linear combinations of atomic orbitals (AO's) on one atom with different orbital angular momentum quantum number l. They can often be used to make bonding orbitals that yield the same Slater determinant as the total wave function as the canonical MO's. AO's are used as basis functions. Calculations are generally simplified when they are adapted to the symmetry of the molecule. This leads to symmetry-adapted MO's.
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