• Next, the system of Hartree–Fock equations is solved;
• The set of new orbitals being the solution of the HF equations is used in the next
cycle of calculations to determine a new Fock operator;
• The system of Hartree–Fock equations with the new Fock operator is solved;
• The last two steps are repeated until the orbitals used to define the Fock operator
and the orbitals obtained as a result of solving the system of HF equations with
this operator (or total energies) differ by less than the predetermined, required
maximum error (i.e., until we get a self-consistent solution).
By applying these two approximations (BOA and IEA) to initial TISE, we have
made a big step toward the practical applicability of QM to real systems. But one
thing remains unresolved—we have assumed electrons to be independent particles,
but we have to define yet the functions describing (spin)orbitals in Slater determinant—the state vectors of these electrons. Since we do not know the exact form
of such functions, we are forced to make some assumptions about their shape and
usually define these functions as a linear combination of some analytical functions
and search for the best form (best expansion coefficients for chosen set of analytical
functions) employing variational principle. In principle, one can choose any
functions, but in practice, due to the numerical efficiency, only limited set of various
analytical functions is used—depending on the system of interest, the functions
which “mimic” the electron distribution of real system best, are chosen, since this
can greatly decrease the number of calculation cycles and thus the time necessary to
carry out the calculations. For molecular system, the usual choice are atomic
orbitals (AO)—one can safely assume that when atoms approach each other and
start to form the molecule, the resulting electron density distribution will change,
but will resemble, even in case of valence electrons, the original atomic distribution
and thus the linear combination of such atomic orbitals, defining the shape of
molecular orbital should be the most efficient one in molecular system. But again—
since the exact form of atomic orbitals is not known, one can further approximate
AO by means of linear combination of some simple analytical functions. Originally
Slater-type orbitals were used, but later, due to their numerical inefficiency, they
were substituted in most quantum chemical codes by Gaussian-type orbitals,
functions poorly describing the density distribution (so we need more of them in
linear combination), but definitely more efficient numerically, which more than
enough compensates their inefficient shape. Such Gaussian-type orbitals can be
single Gaussian functions (primitives) or their linear combination (contracted).
Currently, many predefined basis sets (depending on a definition of atomic orbitals
by means of Gaussian functions) is available, and the user is responsible for a
choice of the basis set (theory level) best suited for the system being studied. In case
of periodic solids, where one can find both extreme cases (metals and molecular
crystals) and a whole lot of intermediate ones, electrons can be described by plane
waves (best suited for metals) or atomic orbitals (best for molecular crystal). One
can use either of them (having in mind that their efficiency will be quite different
depending on the crystal studied) or some kind of hybrid functions like (L)APW,
ASW, (L)ASO, (L)MTO, to name the few—see, e.g., [29] for more details.
1 Computational Methods in Spectroscopy
13
• The set of new orbitals being the solution of the HF equations is used in the next
cycle of calculations to determine a new Fock operator;
• The system of Hartree–Fock equations with the new Fock operator is solved;
• The last two steps are repeated until the orbitals used to define the Fock operator
and the orbitals obtained as a result of solving the system of HF equations with
this operator (or total energies) differ by less than the predetermined, required
maximum error (i.e., until we get a self-consistent solution).
By applying these two approximations (BOA and IEA) to initial TISE, we have
made a big step toward the practical applicability of QM to real systems. But one
thing remains unresolved—we have assumed electrons to be independent particles,
but we have to define yet the functions describing (spin)orbitals in Slater determinant—the state vectors of these electrons. Since we do not know the exact form
of such functions, we are forced to make some assumptions about their shape and
usually define these functions as a linear combination of some analytical functions
and search for the best form (best expansion coefficients for chosen set of analytical
functions) employing variational principle. In principle, one can choose any
functions, but in practice, due to the numerical efficiency, only limited set of various
analytical functions is used—depending on the system of interest, the functions
which “mimic” the electron distribution of real system best, are chosen, since this
can greatly decrease the number of calculation cycles and thus the time necessary to
carry out the calculations. For molecular system, the usual choice are atomic
orbitals (AO)—one can safely assume that when atoms approach each other and
start to form the molecule, the resulting electron density distribution will change,
but will resemble, even in case of valence electrons, the original atomic distribution
and thus the linear combination of such atomic orbitals, defining the shape of
molecular orbital should be the most efficient one in molecular system. But again—
since the exact form of atomic orbitals is not known, one can further approximate
AO by means of linear combination of some simple analytical functions. Originally
Slater-type orbitals were used, but later, due to their numerical inefficiency, they
were substituted in most quantum chemical codes by Gaussian-type orbitals,
functions poorly describing the density distribution (so we need more of them in
linear combination), but definitely more efficient numerically, which more than
enough compensates their inefficient shape. Such Gaussian-type orbitals can be
single Gaussian functions (primitives) or their linear combination (contracted).
Currently, many predefined basis sets (depending on a definition of atomic orbitals
by means of Gaussian functions) is available, and the user is responsible for a
choice of the basis set (theory level) best suited for the system being studied. In case
of periodic solids, where one can find both extreme cases (metals and molecular
crystals) and a whole lot of intermediate ones, electrons can be described by plane
waves (best suited for metals) or atomic orbitals (best for molecular crystal). One
can use either of them (having in mind that their efficiency will be quite different
depending on the crystal studied) or some kind of hybrid functions like (L)APW,
ASW, (L)ASO, (L)MTO, to name the few—see, e.g., [29] for more details.
1 Computational Methods in Spectroscopy
13