Frank L. Pilar Unlverslty of New Hampshre Durham. 03824
4s is Always Above 3d! or, How to tell the orbitals from the wavefunctions
Textbook diagrams showing that the energy o f an electron in a 4s "level" is less than that of an electron in a 3d "level" for atoms of atomic number 19 andabove have no rigorous basis in experiment or in hona fide quantum theory. Many textbooks of general chemistry contain a diagrameither with the chapter on atomicelectronicstructure, or the chanter on the oeriodic table-said to reoresent the e n e r n levels of electrons within an atom. The iiagram invariabi; shows that the energy of an electron in a 4 s "level" is less than in a 3d "level" for atomic numbers of 19 and above. Yet, such a diagram has no rigorous basis either in experiment or in hona fide quantum theory. First, individual electrons in manyelectron atoms do not exist in energy levels-at least not in the sense of stationary quantum states-and second, when orbital energies (as the electron "levels" are properly termed) are correctly defined, the 4s orbital energy is always abore the 3d orhital energy. The chemist's model of the many-electron atom is based on many features of the exact solution of the nonrelativistic Schrijdinger equation for the hydrogen atom. In this special case there is only a single electron moving in the field of the nucleus and the solutions labelled is, 2s, 2p-refer to stationary quantum states of the atom. Unfortunately, chemical educators have been careless in the development of the concepts needed in building up a qualitative model of an atom containine more than one electron: the role laved hv the additionaiklectron(s) is not as trivial and disposihfe as u&ally assumed. In narticular. terms such as orhital. orhital enerrv. -., energy leve1,'and wavefunction either have been misused or employed in rather vague ways. Consequently, although chemists frequently allude to the Schrodinger equation as their ultimate authoritv-and sometimes even disolav . " it in freshmen texts in full'differential attire!-they invariably proceed to develop a model for atomic electronic structure which not only is in variance with various quantum mechanical principles hut contains internal inconsistencies as well. ~ n y b n ewho has tried to teach the rigorous quantum mechanical basis of the orhital approximation to upperclass or beginning graduate chemistry students knows how much has to he unlearned. The wauefunction of an atom or molecule is a mathematical quantity which contains all possible relevant information about the system. Strictly speaking, only an isolatedsystern can he described by a wavefunction. The simplest satisfactory auantum mechanical model of a manv-electron atom assumes t h . ~ tthe elecrrmic wa\.efunction may he q~prwuoilredI n ;I S h t e r detmninant oi one-electron diirriliut~onfunctitm which contain both spatial and spin coordinates of the electrons. For the special case of an atom with an even number of electrons (2N), all in closed shells, the determinantal wavefunction is customarily assumed to have the special form J.l1,2,. . . 2N) =
$l(l)*(l)$1(1)#(1)... $ N ~ I ) I Y$,VI!)[j(l) ~~)
...... ....... $N(zN)~(zN) where 4 i ( ~is) the spatial part of the one-electron distrihution function used to descrihe the behavior of electron F. The 2 1 Journal of Chemical Education
quantities &) and O(@) are the spin portions of the oneelectron distrihution functions.l The determinantal form is required in order to satisfy the important fundamental antisymmetry principle of quantum mechanics.? The physical interpretation of the determinantal wavefunction is that of a superposition of 2N single-electron distrihution functions to serve as an aooroximation to a sinele 2N-electron distri.. hution function. The form of the approximate wavefunction is such that each electron is as likelv to be described hv a particular function $; as by any other; this is necessary in order to avoid imolvine distineuishahilitv of electrons. * At this point, the one-electron distrihution functions-or snecificallv. .. their snatial nortions-are completelv arhitrarv: f h w i n g Mullikenihey a;e given the generainamdoforbitais. Since orbitals describe a sinrle electron under the auerace
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of ti-i atom. Nevertheless, hydrogen atom wavefunctions do not turn out to be eood orbitals for manv-electron atoms.:' Considerable improvement results by modifying the hydrogen wavefunctions by using the nuclear charge Z as an adjustable parameter. One way of interpreting the effect of varyingZ is in terms of a shielding or screening of nuclear charge by the electrons; this is a crude and incomplete way of taking electron reuulsions into account. Nevertheless, most fixed sets of orhiials generally lead to certain internal inconsistencies, 'In general, h r N electrons IN odd or even), alinear combination of two or mare Slater determinants may be needed. However, must uf the qualitative aspects d t h e model do not depend on this gener-
alization. 'Since the exact wavefunction must be antisymmetric with respect to exchange of the full coordinates (space and spin) of any pair uf electrons,appnximate wavefunetims will he all the better if they also obey the antisymmetry principle. Incidentally, the Pauli exclusi~m principle is not a fundamental principle i,f quantum mechanics. It arises only hecause of the particular form la determinantal product) of the approximate wavefunction employed ta satisfy the antisymmetrg principle. :'For example, they are not a complete set of basis functions and they do not lead to satisfaction 01 the quantum mechanical virial theorem. In addition land partlyas o mnsequenee of thelatter) they give very poor approximations to the atom's total energy and do not correlate well with ionization energies and X-ray and spectral term values.
...the total energy o f the atom is not simply the sum of the orbital energies. .. notably, that if a single electron is assumed to have a charge distribution implied by an orhital 4;. then use of this orhital to define an effective Hamiltonian for the electron-one taking into account the average effect of the other 2N - 1 electrk-leads to the implic&on that the electron distribution is,not given by the assumed 4; hut by some modified orhital 4;. It was the genius of the Hartrees (father and son) to suggest a way of ohtainingorhitals free of this inconsistency. Such orbitals are now known as SCF (for self-consistent field) or Hartree-Fock orhitals.' What a t first appears to he an entirely different approach is to use the quantum mechanical variation theorem to select the best orhitals, that is, to choose those orhitals which make the total energy of the atom a minimum. Somewhat surprisingly, the two approachesself-consistencv and enernv minimization-lead to the same total wavefunc&n. ~ h u ~ t orhitals h e obtained by the two methods can alwavs he transformed into each other without affecting the energy or the intrinsic nature of the model. The S C F orbitals can always he chosen to satisfy the pseudoeigenvalue equations known as the Hartree-Fock or self-consistent field equations 34; =
