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3 The Application of the Schrödinger Equation to Chemistry by Hückel

3 The Application of the Schrödinger Equation to Chemistry by Hückel

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From the above we see that we supply a number to a function to get a number,

and we supply a function to a functional to get a number. By analogy, we supply a

functional to a “2-functional” to get a number. I leave a specific example as an

exercise for the reader.

Chapter 7, Harder Questions, Answers


Why is it that the HF Slater determinant is an inexact representation of the

wavefunction, but the DFT determinant for a system of noninteracting electrons

is exact for this particular wavefunction?

The HF (Hartree–Fock) Slater determinant is an inexact representation of the

wavefunction because even with an infinitely big basis set it would not account

fully for electron correlation (it does account exactly for “Pauli repulsion” since if

two electrons had the same spatial and spin coordinates the determinant would

vanish). This is shown by the fact that electron correlation can in principle be

handled fully by expressing the wavefunction as a linear combination of the HF

determinant plus determinants representing all possible promotions of electrons

into virtual orbitals: full configuration interaction. Physically, this mathematical

construction permits the electrons maximum freedom in avoiding one another.

The DFT determinant for a system of noninteracting electrons is exact for this

particular wavefunction (i.e. for the wavefunction of the hypothetical noninteracting electrons) because since the electrons are noninteracting there is no need to

allow them to avoid one another by promotion into virtual orbitals.

For an account of DFT that is at once reasonably detailed, clear and concise see

Cramer [1].


1. Cramer CJ (2004) Essentials of computational chemistry. Wiley, Chichester, England, chapter 8

Chapter 7, Harder Questions, Answers


Why do we expect the “unknown” term in the energy equation (Exc[r0], in

Eq. 7.21) to be small?



Eq. 7.21 is



E0 ¼


nuclei A




r0 r1 ị


2 KS



1 1ịjr1 jc1 1ị


2 iẳ1

r0 r1 ịr0 r2 ị

dr1 dr2 ỵ EXC ẵr0


Exc[r0] is a correction term to the electronic kinetic and potential energy; most

of this energy is (we hope!) treated classically by the other terms [1].


1. Cramer CJ (2004) Essentials of computational chemistry. Wiley, Chichester, England, sections 8.3 and 8.4

Chapter 7, Harder Questions, Answers


Merrill et al. have said the “while solutions to the [HF equations] may be viewed as

exact solutions to an approximate description, the [KS equations] are approximations to an exact description!” Explain.

Solutions to the Hartree–Fock equations are exact solutions to an approximate

description because:

The HF equations are approximate mainly because they treat electron–electron

repulsion approximately (other approximations are mentioned in the answer suggested for Chapter 5, Harder Question 1). This repulsion is approximated as

resulting from interaction between two charge clouds rather than correctly, as the

force between each pair of point-charge electrons. The equations become more

exact as one increases the number of determinants representing the wavefunctions

(as well as the size of the basis set), but this takes us into post-Hartree–Fock

equations. Solutions to the HF equations are exact because the mathematics of

the solution method is rigorous: successive iterations (the SCF method) approach an

exact solution (within the limits of the finite basis set) to the equations, i.e. an exact

value of the (approximate!) wavefunction CHF.

The Kohn–Sham equations are approximations because the exact functional

needed to transform the electron density function r into the energy is unknown.

They are approximations to an exact description because the equations (as distinct

from methods of solving them) involve no approximations, with the ominous caveat

that the form of the r-to-E functional Exc is left unspecified.



Chapter 7, Harder Questions, Answers


Electronegativity is the ability of an atom or molecule to attract electrons. Why is it

then (from one definition) the average of the ionization energy and the electron

affinity (Eq. 7.32), rather than simply the electron affinity?

Equation 7.32 is




We can call this the Mulliken electronegativity. Why is electronegativity not

defined simply as the electron affinity (A)? First, we saw two derivations of

Eq. 7.32. In the first, electronegativity(w) was intuitively taken as the negative of

electronic chemical potential (the more electronegative a species, the more its

energy should drop when it acquires electrons). This led to approximating the

derivative of energy with respect to number of electrons at a point corresponding

to a species M as the energy difference of M+ and MÀ divided by 2. In the second,

Mulliken, derivation, a simple argument equated electron transfer from X to Y to

transfer from Y to X. Both derivations clearly invoke ionization energy (I). It is no

surprise that w should be connected with A, but the intrusion of I may be puzzling;

however, our surprise diminishes if we note that the more electronegative a species,

the more readily it should gain an electron and the less readily it should part

with one.

But could we alternatively reasonably define electronegativity quantitatively just

as electron affinity? Let’s compare with the popular Pauling electronegativity scale

[1] electronegativities calculated from Eq. 7.32 and calculated simply as A. (The

Pauling scale has been criticised by Murphy et al. [2] and their criticisms were

acknowledged and improvements to the scale suggested, by Smith [3]; Matsunaga

et al., provided a long defence of Pauling’s scale [4]). Below are some electronegativities (preceded by a table of the calculated needed energies, at the MP2/6311ỵG* level) by these three methods.

Energies in hartrees












Starting from a neutral quintet 1s2, 2s1, 2px1, 2py1, 2pz1

Starting from a neutral triplet 1s2, 2s2, 2px1, 2py1, 2pz0











I, A, and Mullikenw, in eV, Pauling w in kJ molÀ1. Hartrees were converted to eV

by multiplying by 27.212.



I and A were calculated as the energy difference between the neutral and the

cation and anion, respectively.













Pauling w







Starting from a neutral quintet 1s , 2s , 2px , 2py1, 2pz1


Starting from a neutral triplet 1s2, 2s2, 2px1, 2py1, 2pz0


Based on experimental bond energies in C–X molecules











We see that the Mulliken and Pauling electronegativities seem to be reasonably

in step, with electronegativity increasing from Li to C to F, in accord with experience, but with A making quintet C more electronegative than F. Evidently I and A

act together to determine atomic avidity for electrons.

Electronegativity and other properties from DFT calculations have been discussed by Zhan et al. [5], and an electronegativity scale based on the energies of

neutrals and cations which correlates well with the Pauling scale has been proposed

by Noorizadeh and Shakerzadeh [6].


1. Pauling L (1932) J Am Chem Soc 54:3570; Pauling L (1960) The nature of the chemical bond,

3rd edn. Cornell University Press, Ithaca, NY, chapter 3

2. Murphy LR, Meek TL, Allred AL, Allen IC (2000) J Phys Chem A 104:5867

3. Smith DW (2002) J Phys Chem A 106:5951

4. Matsunaga N, Rogers DW, Zavitsas AA (2003) J Org Chem 68:3158

5. Zhan C-G, Nichols JA, Dixon DA (2003) J Phys Chem A 107:4184

6. Noorizadeh S, Shakerzadeh E (2008) J Phys Chem A 112:3486

Chapter 7, Harder Questions, Answers


Given the wavefunction of a molecule, it is possible to calculate the electron density

function. Is it possible in principle to go in the other direction? Why or why not?

From density functional theory, given the electron density function of a molecule (and its charge and multiplicity), and a perfect functional (let’s idealize the

problem; the question does specify “in principle”) we can home in on a unique

molecule. Then we could use ab initio theory to find the wavefunction.



Chapter 7, Harder Questions, Answers


The multielectron wavefunction C is a function of the spatial and spin coordinates

of all the electrons. Physicists say that C for any system tells us all that can be

known about the system. Do you think the electron density function r tells us

everything that can be known about a system? Why or why not?

Although the wavefunction C seems to contain more information than the

electron density function r (question 1), it ought to be possible in principle to

calculate any property of a system from r, because different states – different

geometries, different electronic states, etc. – must have different electron distributions (or they would not be different). The problem is to transform the calculated

r to an energy (question 5).

Extraction of information from r may not be as elegant as from C. For example,

the Woodward–Hoffmann rules follow fairly transparently from the symmetries of

molecular orbitals (wavefunctions), but deriving them from r requires using a “dual

descriptor” function [1].


1. Ayers PW, Morell C, De Proft F, Geerlings P (2007) Chem Eur J 13:8240

Chapter 7, Harder Questions, Answers


If the electron density function is mathematically and conceptually simpler than the

wavefunction concept, why did DFT come later than wavefunction theory?

The wavefunction [1] and electron density [2] concepts came at about the same

time, 1926, but the application of wavefunction theory to chemistry began in the

1920s [3], while DFT was not widely used in chemistry until the 1980s (see below).


The DFT concept of calculating the energy of a system from its electron density

seems to have arisen in the 1920s with work by Fermi, Dirac, and Thomas.

However, this early work was useless for molecular studies, because it predicted

molecules to be unstable toward dissociation. Much better for chemical work, but

still used mainly for atoms and in solid-state physics, was the Xa method, introduced by Slater in 1951. Nowadays the standard DFT methodology used by

chemists is based on the Hohenberg–Kohn theorems and the Kohn–Sham approach



for implementing them (1964, 1965). It is not far from the truth to say that the use of

DFT in chemistry began, with this method, in the 1960s. The first such calculation

was on atoms (1966) [4], with molecular DFT calculations picking up steam in the

1970s [5], and starting to become routine ca. 1990 [6].

The reason for the delay is that it took the Kohn–Sham approach to initiate

practical DFT calculations on molecules, and time was needed to “experiment”

with techniques for improving the accuracy of calculations [7]. As for why the

Hohenberg–Kohn theorems and the Kohn–Sham insight came not until 40 years

after the wavefunction and electron density concepts, one can only speculate;

perhaps scientists were mesmerized by the peculiarities of the wavefunction [8],

or perhaps it simply took the creativity of specific individuals to usher in the era of

widespread density functional calculations.


1. The Schr€odinger equation applied the wave concept of particles to a classical wave equation

yielding wavefunctions as solutions: Schr€

odinger E (1926) Ann Phys 81:109

2. The interpretation of the square of the wavefunction as a measure of electron density in atoms

and molecules arose from a slightly different suggestion by Max Born: Born M (1926) Z Phys

37:863. See Moore W (1989) Schr€

odinger. Life and thought. Cambridge University Press,

Cambridge, UK, pp 219–220, 225–226, 240, 436–436

3. Both the early molecular orbital and the early valence bond approaches used wavefunctions: (a)

Molecular orbital, e.g. Pauling L (1928) Chem Rev 5:173. Lennard-Jones E (1929) Trans

Faraday Soc 25:668. (b) Valence bond: Heitler W, London F (1927) Z Phys 44:455

4. Tong BY, Sham LJ (1966) Phys Rev 144:1

5. A search of Chemical Abstracts with SciFinder using the article title words “density functional”

gave for 1950–1970, only one publication, but for 1971–1979, 111 publications, and for 1980,

45 publications

6. Borman S (1990) Chemical and Engineering News, April 9, p 22

7. For a short exposition of the evolution from the local-density approximation to the local-spindensity approximation and gradient-corrected and hybrid functionals, see Levine IN (2000)

Quantum chemistry, 5th edn. Prentice Hall, Upper Saddle River, NJ, pp 581–592

8. E.g. (a) Baggott J (1992) The meaning of quantum theory. Oxford Science Publications,

Oxford. (b) Whitaker A (1996) Einstein, Bohr and the quantum dilemma. Cambridge University Press, Cambridge, UK

Chapter 7, Harder Questions, Answers


For a spring or a covalent bond, the concepts of force and force constant can be

expressed in terms of first and second derivatives of energy with respect to

extension. If we let a “charge space” N represent the real space of extension of

the spring or bond, what are the analogous concepts to force and force constant?

Using the SI, derive the units of electronegativity and of hardness.



Force and force constant, for a spring or bond, reflect the dependence of energy

on extension:

Force ¼ F ¼ ÀdE=dx


Force constant ¼ k ¼ ÀdF=dx ¼ d2 E=dx2


(Force is a vector, acting in the opposite direction to the that along which the

spring or bond is extended, hence the minus sign; the force constant is positive).

Energy and charge density are closely connected, E being a functional of r for the

ground state:

E0 ẳ Fẵr0


We want equations analogous to Eqs. 1 and 2 with r instead of E. Equation 3

leads us to

Force ẳ F ẳ dFẵr=dx


Force constant ẳ k ẳ dF=dx ẳ d2 Fẵr0 =dx2



both for the ground electronic state.

Units of electronegativity and hardness in the international system

Electronegativity can be defined as



w ¼ Àm ¼ À

@N V


and hardness can be defined as


@N 2











Within these definitions, the units of electronegativity must then be

Change in energy/change in pure number ¼ J (joules)

and the units of hardness must be

Change in electronegativity/change in pure number

¼ change in J/change in pure number ¼ J

Electronegativity is a measure of how fast energy changes as electrons are

added, and hardness is a measure of how fast electronegativity changes as electrons

are added. In the “classical” Pauling definition, electronegativity is commonly said

to be dimensionless, but should really have the units of square root of energy

(arising from bond energy difference to the power of 1/2), and in the Mulliken

definition the units are those of energy (see Chapter 7, Harder Question 6).



Chapter 8, Harder Questions, Answers


1. In microsolvation, should the solvent molecules be subjected to geometry


Ideally, the solvent molecules, as well as the solute molecules, should be

subjected to geometry optimization in microsolvation (implicit solvation): in a

perfect calculation all components of the system, in this case the solution, would

be handled exactly. This is feasible for most quantum mechanical (AM1 or PM3,

ab initio, DFT) microsolvation calculations, since these usually use only a few

solvent molecules (see e.g. Chapter 8, [14]). Forcefield (molecular mechanics)

calculations on biopolymers surround the solute with a large number of molecules when implicit solvation is used, and it may not be practical to optimize


2. Consider the possibility of microsolvation computations with spherical, polarizable “pseudomolecules”. What might be the advantages and disadvantages of

this simplified geometry?

The advantages come from geometric simplicity: the orientation of the molecules with respect to the solute does not have to be optimized, nor does the more

ambitious task of solute molecule geometry optimization arise.

The disadvantages stem from the fact that the only solvents that really consist of

spherical molecules are the noble gases. These are used as solvents only in quite

specialized experiments, for example:

(a) Solvent effect on the blue shifted weakly H-bound F3CHÁÁÁFCD3 complex:

Rutkowski KS, Melikova SM, Rodziewicz P, Herrebout WA, van der Veken

BJ, Koll A (2008) J Mol Struct 880:64

(b) Liquid noble gases as ideal transparent solvents: Andrea RR, Luyten H,

Stufkens DJ, Oskam A (1986) Chemisch Magazine (Den Haag), (January)

23, 25. (In Dutch)

(c) Depolarization of fluorescence of polyatomic molecules in noble gas solvents: Blokhin AP, Gelin MF, Kalosha I, Matylitsky VV, Erohin NP,

Barashkov MV, Tolkachev VA (2001) Chem Phys 272:69

3. In microsolvation, why might just one solvent layer be inadequate?

The essential reason why one (or probably two or three) solvent layers is

not enough is that with, say, one layer the solvent molecules in contact with a

solute molecule are not “distracted’ by an outer layer and so turn their solvating

power on the solute more strongly than if they also had to interact with an outer

solvent layer (see Bachrach SM (2007) Computational organic chemistry.

Wiley-Interscience, Hoboken, NJ, chapter 6). The solute is evidently oversolvated. Formally, we can say that n layers is sufficient if going to n ỵ 1 layers has

no significant effect on the phenomenon we are studying. Unfortunately, it is not

yet possible to computationally find this limiting value of n for higher-level

quantum mechanical calculations.



4. Why is parameterizing a continuum solvent model with the conventional dielectric constant possibly physically unrealistic?

The conventional dielectric constant is an experimental quantify that refers to

the solvent as a continuous insulating medium. On the molecular scale solute

and solvent are not separated by a smooth medium, but rather by discrete

particles (molecules) with empty interstices.

5. Consider the possibility of parameterizing a continuum solvent model with

dipole moment.

Continuum solvent models are normally parameterized with the solvent dielectric

constant (but see the COSMO models, Chapter 8). First we note that dielectric

constant and dipole moment are not in general well correlated; from Chapter 8:

For 24 solvents encompassing nonpolar (e.g. pentane, m 0.00, e 1.8), polar aprotic (e.g.

dimethyl sulfoxide, m 3.96, e 46.7), and polar protic (e.g. water, m 1.85 e 80) dispositions,

the correlation coefficient r2 of e with m was only 0.36 (removing formic acid and water

raised it to 0.75). For nine nonpolar, seven polar aprotic, and 8 polar protic solvents,

considered as separate classes, r2 was 0.90, 0.87, and 0.0009 (sic), respectively .....

If we consider just essentially using dipole moment as a surrogate for dielectric

constant, with minor conceptual adjustments like some changes in the parameterization constants, then from the above, for nonpolar and polar aprotic solvents the

correlation is good enough that it may be possible to parameterize with dipole

moment, but there is no clear indication that this would have any advantage.

Furthermore, water, the most important solvent, belongs to the polar protic class,

for which there is no correlation.

Less clear is whether a different approach than that used with dielectric constant

might be fruitful with dipole moment. A useful solvation algorithm does not seem to

have emerged from studies of the effect of dipole moment on solvation energies, e.g.

(a) Effect of bond and group dipole moments on the enthalpy of solvation of

organic nonelectrolytes: Antipin IS, Karimova LKh, Konovalov AI (1990) Z

Obshch Khim 60:2437–2440. (In Russian)

(b) Free energy of solvation of aromatic compounds and their polarizability:

Gorbachuk VV, Smirnov SA, Solomonov BN, Konovalov AI (1988) Dokl

Akad Nauk SSSR 300:1167. This paper studied dipole moment as well as

polarizability. (In Russian)

Chapter 8, Harder Questions, Answers

Singlet Diradicals

1. Is CASSCF size-consistent?

We saw that full CI is size-consistent (Section 5.4.3). Now, CASSCF is complete

CI, within a specified set of molecular orbitals. If done right it is size-consistent.



Done right means that in comparing the energy of two systems one must utilize

corresponding electron promotions (“excitations”). I’ll illustrate this by comparing the energy of two well-separated beryllium atoms with the twice the energy

of one beryllium atom. I choose the beryllium atom because this 4-electron atom

is the simplest closed-shell species which gives some choice (among 2s and 2p

set) of occupied orbitals, lending a little resemblance in this respect to the

molecular case.

A CASSCF(2,2)/6-31G* calculation was done on one beryllium atom, using a

simplified version of the procedure in Chapter 8 for molecules: an orbital

localization step is pointless for an atom, and in the energy calculation optimization is meaningless. First an STO-3G wavefunction was obtained and the

atomic orbitals (AOs) were visualized; this showed MO1, 2, 3, 4, and 5 to be,

respectively, 1s, 2s (both occupied), and three energetically degenerate unoccupied 2p orbitals. The active space was chosen to consist of the 2s and a 2p

orbital, and a single-point (no optimization requested) CASSCF(2,2)/6-31G*

calculation was done. The energy was À14.5854725 hartree.

A CASSCF(2,2)/6-31G* calculation was now done on two beryllium atoms

˚ , where they should be essentially noninteracting; the coordiseparated by 20 A

nates of these two atoms were input treating them as one unit, an 8-electron

supermolecule. An STO-3G wavefunction was obtained and visualized. This

showed as expected a set of molecular orbitals (MOs), since this species is

formally a molecule. With five AOs from each atom, we have ten AOs resulting

from plus and minus combinations (bonding and antibonding only in a formal

sense, because of the separation). These were:

MO1, 1s ỵ 1s; MO2, 1s-1s; same energy. These two account for two pairs of


MO3, 2s þ 2s; MO4, 2s-2s; same energy. These two account for two pairs of


MO5, 2px ỵ 2px; MO6, 2px 2px; ...., MO10, 2pz À 2pz, All six of these, 5–10,

same energy, unoccupied.

The critical choice was made of a CASSCF(4,4)/6-31G* calculation; the active

space is thus the degenerate filled 2s ỵ 2s and 2s À 2s pair of MOs, and the

degenerate empty 2px ỵ 2px and 2px 2px pair of MOs. CASSCF(4,4) was

chosen because it corresponds to the CASSCF(2,2) calculation on one beryllium

atom in the sense that we are doubling up the number of electrons and orbitals in

our noninteracting system. This calculation gave an energy of À29.1709451

hartree. We can compare this with twice the energy of one beryllium atom, 2 Â

À14.5854725 hartree ¼ À29.1709450 hartree.

Let’s compare these CASSCF results with those for a method that is not sizeconsistent, CI with no “complete” aspect. We’ll use CISD (configuration interaction singles and doubles; Section 5.4.3). Here are the results for CISD/6-31G*:

One beryllium atom, À14.6134355

˚ , À29.2192481.

Two beryllium atoms separated by 20 A



This is significantly higher than with twice the energy of one beryllium atom:

2 Â À14.6134355 ¼ À29.226871; À29.2192481 À (À29.226871) ¼ 0.00762

hartree or 20.0 kJ molÀ1. If unaware that CISD is not size-consistent, one might

have thought that these widely-separated atoms are destabilized by 20 kJ molÀ1.

By comparison, the hydrogen-bonded (stabilizing) enthalpy of the water dimer

lies in the range 13–21 kJ molÀ1 (Chapter 5, reference [104]).

2. In one-determinant HF (i.e. SCF) theory, each MO has a unique energy

(eigenvalue), but this is not so for the active MOs of a CASSCF calculation.


The MOs used for the active space are normally localized MOs, derived from the

canonical MOs (Section by taking linear combinations of the original

MOs of the Slater determinant. Localization has no physical consequences: C

expressed as the “localized determinant” is in effect the same as C expressed as

the canonical determinant, and properties calculated from the two are identical.

However, the canonical MOs and the localized MOs are not the same: in the two

sets of MOs the coefficients of the basis functions are different, which is why

canonical and localized MOs look different. Each canonical MO has an eigenvalue which is approximately the negative of its ionization energy (Koopmans’

theorem); MO coefficients and eigenvalues are corresponding columns and

diagonal elements of the C and « matrices in Eqs. 4.60 and 5.1. Since the

localized MOs differ mathematically from the canonical, there is no reason

why they should have physically meaningful eigenvalues.

3. In doubtful cases, the orbitals really needed for a CASSCF calculation can

sometimes be ascertained by examining the occupation numbers of the active

MOs. Look up this term for a CASSCF orbital.

In its most general physical use, occupation number is an integer denoting the

number of particles that can occupy a well-defined physical state. For fermions it

is 0 or 1, and for bosons it is any integer. This is because only zero or one

fermion(s), such as an electron, can be in the state defined by a specified set of

quantum numbers, while a boson, such as a photon, is not so constrained (the

Pauli exclusion principle applies to fermions, but not to bosons). In chemistry

the occupation number of an orbital is, in general, the number of electrons in it.

In MO theory this can be fractional.

In CASSCF the occupation number of the active space MO number i (ci) is

defined as (e.g. Cramer CJ (2004) Essentials of computational chemistry, 2nd

edn. Wiley, Chichester, UK, p 206):

occ numb of MOi ẳ



occ numb ịi;n a2n


i.e. it is the sum, over all n configuration state functions (CSFs) containing MOi, of

the product of the occupation number of a CSF and the fractional contribution (a2)

of the CSF to the total wavefunction C. A CSF is the same as a determinant for

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