Rogers Computational Chemistry Using the PC

234 COMPUTATIONAL CHEMISTRY USING THE PC

Solution 8-2

ð1

S11 ¼ xð1 xÞxð1 xÞdt

0

ð1

¼ðx2 x3 x3 þ x4Þdt

Now, calculating all Hij and Sij elements in the same way, and inserting them into

This can be cleared of fractions by multiplying by 1260 to obtain the secular determinant

corresponding to Eq. set (1-9), which was solved iteratively in Computer Project 1- 2 to yield the roots

The lower of the two roots is the one we seek for the ground-state energy of the

as contrasted to the exact solution of 0:125ðh2=mÞ. Note that the energy obtained from the variational solution is slightly higher than the solution obtained from the

exact wave function. [This result makes it clear why we needed to find the roots of Eq. (1-9) to six significant figures.]

One of the things illustrated by this calculation is that a surprisingly good approximation to the eigenvalue can often be obtained from a combination of approximate functions that does not represent the exact eigenfunction very closely. Eigenvalues are not very sensitive to the eigenfunctions. This is one reason why the LCAO approximation and Huckel theory in particular work as well as they do.

Another feature of advanced molecular orbital calculations that we can anticipate from this simple example is that calculating matrix elements for real molecules can be a formidable task.

The Helium Atom

The helium atom is similar to the hydrogen atom with the critical difference that there are two electrons moving in the potential field of a nucleus with a double positive charge ðZ ¼ 2Þ (Fig. 8-1).

we have two Hamiltonians that are identical to the hydrogen case except for a different nuclear charge, plus an added term 1=r12 due to electrostatic repulsion of the two electrons acting over the interelectronic distance r12

from the single electron or orbital approximation, assuming that the orbital of helium is separable into two one-electron orbitals ð1; 2Þ ¼ cð1Þcð2Þ.

The kinetic energy operator for the one-electron system of the H atom is ðh2=2mÞð1=r2Þðd=drÞr2ðd=drÞ [Eq. (6-21)] and the potential energy is e2=r for attraction of a single electron to the hydrogen nucleus. It is reasonable to use the same operator for a single electron in a separated helium orbital, either cð1Þ or cð2Þ. In atomic units we have

for each kinetic energy part and 2=r as each potential energy part.

Although we are solving for one-electron orbitals, c1 and c2, we do not want to fall into the trap of the last calculation. We shall include an extra potential energy term V1 to account for the repulsion between the negative charge on the first electron we consider, electron 1, exerted by the other electron in helium, electron 2. We don’t know where electron 2 is, so we must integrate over all possible locations of electron 2

The same treatment produces a similar operator for electron 2.

We do not know the orbitals of the electrons either. (An orbital, by the way, is not a ball of fuzz, it is a mathematical function.) We can reasonably assume that the ground-state orbitals of electrons 1 and 2 are similar but not identical to the 1s orbital of hydrogen. The Slater-type orbitals

240 COMPUTATIONAL CHEMISTRY USING THE PC

Solution 8-4

The error, on the second iteration, has been reduced to 6.5%. Notice that the calculated IP1 on this iteration is too large.

COMPUTER PROJECT 8-1 j The SCF Energies of First Row Atoms and Ions

Using as many methods as are available to you for comparison (Mathcad, QBASIC, and TRUE BASIC), determine the self-consistent field (SCF) energies of the He atom and of the ions Liþ, Be2þ, and B3þ. Fill in the SCF column of Table 8-1.

Plot the total SCF energy of the He atom and the three two-electron ions as a function of the nuclear charge Z. Describe the curve (linear, monotonic, etc.) so obtained. Using the GAUSSIAN# package, calculate the energies of the atoms and ions in Table 8-1 at the STO-2G and STO-3G levels. The STO-xG levels of calculation are carried out by using a wave function that is the sum of two or more Gaussian functions with parameters chosen so as to approximate a Slater-type orbital. The STO-xG levels of calculation will be discussed in more detail in the next section, but for now, all you need to know is that the six-line input file

# STO-2G

Helium

0 1 He

will provide the solutions for this project. GAUSSIAN for WINDOWS has a template that facilitates writing input files, but input files can also be written using a DOS editor. Mind the blank lines 2 and 4.

To fill out Table 8-1, change the element symbols in the last line to Li, Be, or B and designate the charge and spin multiplicities as 1 1, 2 1, 3 1 in that order. In line 5, the first number is the single positive charge and the second number is the spin multiplicity, 1 for paired electronic spins and 2 for an unpaired electron. A

Table 8-1 Energies (in hartrees) of First Row Atoms and Ions

% error

space between the charge and spin multiplicity is essential. The spin multiplicities are all 1 here because each problem is a two-electron problem and the two spins are paired. To go from an STO-2G calculation to an STO-3G calculation, change the 2 to a 3 in the route section (line 1) of the input file.

COMPUTER PROJECT 8-2 j A High-Level ab initio Calculation of SCF First IPs of the First Row Atoms

In contrast to the low-level calculations using the STO-3G basis set, very high level calculations can be carried out on atoms by using the Complete Basis Set-4 (CBS-4) procedure of Petersson et al. (1991,1994). For atoms more complicated than H or He, the first ionization potential (IP1) calculation is a many-electron calculation in which we calculate the total energy of an atom and its monopositive ion and determine the IP of the first ionization reaction

A ! Aþ þ e

from the difference ðEAþ EAÞ. The CBS-4 ‘‘program’’ is actually a suite of several programs and corrections that are linked to one another so that they are carried out sequentially. The procedure is intended to come very close to the result that would have been obtained by using a complete basis set by extrapolation from a large but (obviously) finite basis set.

Procedure. To go from an STO-3G calculation to a CBS-4 calculation, simply replace STO-3G with CBS-4 in the route section of the program used in Computer Project 8-1. Complete Table 8-2 by filling in the CBS-4 Energies of the atoms and ions listed in columns 1 and 3 of Table 8-2 and put them into columns 2 and 4 of the table. You will notice that some of the simpler atoms (H through Be) do not have a listed CBS-4 Energies, but they do have an SCF energy, which should be used in its place. Calculate the IP and complete column 5. Pay special attention to spin multiplicity and Hund’s rule. The spin multiplicity is n þ 1 where n is the number

Table 8-2 Electronic Energies of Atoms and Single-Positive Ions in the First Row of the Atomic Table

242 COMPUTATIONAL CHEMISTRY USING THE PC

of unpaired electrons in the atom. This may require some thought for atoms C through Ne. If necessary, review the subject of ionization potentials in a good general chemistry textbook (e.g., Ebbing, and Gammon, 1999). Experimental values for column 6 can be obtained from most general chemistry textbooks. They may require unit conversion.

Look up the experimental values of the first ionization potential for these atoms and calculate the average difference between experiment and the computed values. Depending on the source of your experimental data, the arithmetic mean difference should be within 0.010 hartrees. Serious departures from this level of agreement may indicate that you have one or more of your spin multiplicities wrong.

Plot the calculated first IPs as a function of the atomic number Z for the elements from H to Ne in the atomic table. The plot has a characteristic shape that should be familiar from earlier courses. These plots are frequently given in the experimental units of electron volts (eV; hartrees 27:21 ¼ eV) or kilojoules per mole (kJ mol 1; hartrees 2625 ¼ kJ mol 1). Write a paragraph or two in your project report explaining why the graph of IP vs. Z appears as it does.

The STO-xG Basis Set

The true value of for a many-electron atom or a molecule is unknown. If we could set it equal (‘‘expand’’ it) to a linear combination of an infinite number of basis functions, each defined in a space of infinite dimensions, we could carry out an exact calculation of . Such a set of basis functions would be a complete set.

The various basis sets used in a calculation of the H and S integrals for a system are attempts to obtain a basis set that is as close as possible to a complete set but to stay within practical limits set by the speed and memory of contemporary computers. One immediately notices that the enterprise is directly dependent on the capabilities of available computers, which have become more powerful over the past several decades. The size and complexity of basis sets in common use have increased accordingly. Whatever basis set we choose, however, we are attempting to strike a balance. If the basis set is too small, it is inaccurate; if it is too large, it exceeds the capabilities of our computer. Whether our basis set is large or small, if we attempt to calculate all the H and S integrals in the secular matrix without any infusion of empirical information, the procedure is described as ab initio.

Basis functions are themselves contractions of simpler functions called primitives. Contractions are used because they are easier for the computer to handle; hence, they economize on computer power, with the obvious advantage that larger problems can be solved with greater accuracy. We shall illustrate this idea by contracting three Gaussian primitives to approximate a Slater-type orbital (STO). The resulting contraction is called an STO-3G basis function. If this basis is used to describe the 1s atomic orbitals of H and He, it is the minimal basis because H and He have only 1s electrons in the ground state. A minimal basis set contains the smallest number of basis functions necessary for each atom. The minimal basis set for C is 1s, 2s, and 2p. This is larger than the minimal basis set for H because C has a more complicated electronic structure than H. In some approximations, the 1s

electrons of carbon are considered part of the ‘‘core’’ along with the nucleus, and only the 2s and 2p electrons are included in the minimal basis set. Notice that the number of Gaussians does not determine whether a basis set is minimal; both STO2G and STO-3G are minimal. In the use of a contracted basis set, the primitives are not manipulated independently; they form a single basis function and are treated by the computer as a unit.

The Hydrogen Atom: An STO-1G ‘‘Basis Set’’

We shall construct the simplest possible basis function by fitting a single Gaussian to the 1s STO for the hydrogen atom, with the intention of building up to the STO3G basis set later. The task is similar to what was done in Computer Project 6-1. In Part A of that project, we optimized the hydrogen 1s orbital f ða; rÞ ¼ e ar by the variational method and got the exact value, 0.5000 hartrees, for the ground-state energy. In Part B, we found that the lowest variational energy of the Gaussian function gðg; rÞ ¼ e gr2 is obtained when g has been optimized to about 0.83. The result, 0.424 hartrees, is the lowest variational energy you can get from this function, but it is not as good as the result found when the true hydrogenic orbital was used.

In the following, we shall examine the approximation to the Slater-type 1s hydrogen wave function by one Gaussian function using Program GAUSSIAN94W#, a commercial package for Gaussian and related calculations, specifically adapted for a Windows# environment.

The Slater-type orbitals are a family of functions that give us an economical way of approximating various atomic orbitals (which, for atoms other than hydrogen, we don’t know anyway) in a single relatively simple form. For the general case, STOs are written

but for the spherically symmetric 1s orbital of hydrogen, variation in the spherical harmonic Ylmðy; fÞ drops out and n 1 ¼ 0, so

where N is a constant. This function is really just the exact 1s wave function for hydrogen ðrÞ ¼ Ne r because ¼ Z ¼ 1 for this special case, that is, the STO is the same as the wave function .

The Gaussian function can be written

but for the spherically symmetric 1s case, the Cartesian terms xmynzl ¼ 1:0 so we have

where C is a constant.