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Supplement 14-A
The Hartree Approximation
The energy eigenvalue problem for an atom with Z electrons has the form
(14A-1)
and is a partial differential equation in 3 Z dimensions. For light atoms it is possible to solve such an equation on a computer, but such solutions are only meaningful to the ex- pert. We shall base our discussion of atomic structure on a different approach. As in the example of helium ( Z 2), it is both practical and enlightening to treat the problem as one involving Z independent electrons in a single potential, and to consider the electron–electron interaction later. Perturbation theory turned out to be adequate for Z 2, but as the number of electrons increases, the shielding effects, not taken into ac- count by first-order perturbation theory, become more and more important. The varia- tional principle discussed at the end of Chapter 14 had the virtue of maintaining the single-particle picture, while at the same time yielding single-particle functions that incor- porate the screening corrections. To apply the variational principle, let us assume that the trial wave function is of the form
(14A-2)
Each of the functions is normalized to unity. If we calculate the expectation value of H in this state, we obtain
(14A-3)
The procedure of the variational principle is to pick the i ( r i ) such that H is a minimum. If we were to choose the i ( r j ) to be hydrogenlike wave functions, with a different Zi for each electron (and with each electron in a different quantum state to satisfy the Pauli ex- clusion principle), we would get a set of equations analogous to (14-47) and (14-48). A more general approach is that due to Hartree. If the i ( r i ) were the single-particle wave functions that minimized H , then an alteration in these functions by an infinitesimal amount (14A-4) should only change H by a term of order ^2. The alterations must be such that
d^^3 r i i ( r i )^ ^ fi ( r i )^2 ^1
i ( r i ) l i ( r i ) fi ( r i )
H
Z i 1
d^^3 ri * i ( r i )^
^2
2 me ^2 i Ze
2
4 0 ri
i ( r i ) e^
2
4 0 i j j
i ( r i )^2 j ( r j )^2 r i r j
( r 1 , r 2 ,... , r Z ) 1 ( r 1 ) 2 ( r 2 ) …^ Z ( r Z )
E ( r 1 , r 2 ,... , r Z )
Z
i 1 ^
p^2 i 2 me
Ze^2
4 0 ri ^
i j
j
e^2 4 0 r i r j
( r 1 , r 2 ,... , r Z )
W-
that is, to first order in ,
(14A-5)
Let us compute the terms linear in that arise when (14A-4) is substituted into (14A-3). Term by term, we have
(14A-6)
To obtain this we have integrated by parts two times, and used the fact that fi ( r i ) must van- ish at infinity in order to be an acceptable variation of a square integrable function. Next we have
(14A-7)
and finally
(14A-8)
We cannot just set the sum of these three terms equal to zero because the fi ( r i ) are con- strained by (14A-5). The proper way to account for the constraint is by the use of La- grange multipliers; that is, we multiply each of the constraining relations (14A-5) by a constant (the “multiplier”) and add the sum to our three terms. The total can then be set equal to zero, since the constraints on the fi ( r i ) are now taken care of. With a certain amount of notational foresight we label the multipliers i , and thus get
(14A-9)
In deriving the second line, first we converted the double sum i j j into (1/2) i j j , which is unrestricted except for the requirement that i j , and then used the fact that the integrand in (14A-8) is symmetric in i and j. Now fi ( r i ) is completely unrestricted, so that we may treat fi ( r i ) and as completely independent (each one has a real and an imag- inary part). Furthermore, other than being square integrable, they are completely arbitrary, so that for (14A-5) to hold, the coefficients of fi ( r i ) and must separately vanish at each point r i , since we are allowed to make local variations in the functions fi ( r i ) and We are thus led to the condition that
(14A-10)
and the complex conjugate relation.
2 2 me ^2 i Ze
2 4 0 ri e
2
4 0 j i^ ^
d^3 r j
j ( r j )^2 r i r j
i ( r i ) ii ( r i )
f * i ( r i ).
f * i ( r i )
f * i ( r i )
i d^3 r i f * i ( r i ) i ( r i ) complex conjugate terms 0
e
2
4 0 i j j
d^3 r i d^3 r j f * i ( r i )
j ( r j )^2 r i r j
i ( r i )
i
d^3 r i f * i ( r i )
2 2 m ^2 i i ( r i ) Ze
2 4 0 ri i ( r i )
e^2
4 0 i j j^ ^
d^3 r i d^3 r j^1
r i r j
[ f * i ( r i ) i ( r i ) fi ( r i ) * i ( r i )] j ( r j )^2 ( i i j )
i
d^3 ri f * i ( r i ) Ze
2 4 0 ri i ( r i ) * i ( r i ) Ze
2 4 0 ri fi ( r i )
i
d^3 r i fi ( r i )
2 2 m ^2 i * i ( r i ) f * i ( r i )
2 2 m ^2 i i ( r i )
i
d^3 ri * i ( r i )
2 2 m ^2 i (^) fi ( r i ) f * i ( r i )
2 2 m ^2 i (^) i ( r i )
d^3 r i [ * i ( r i ) fi ( r i ) i ( r i ) f * i ( r i )] 0
W-52 Supplement 14-A The Hartree Approximation
The number of electrons that can be placed in orbitals with a given ( n , l ) is 2(2 l 1), since there are two spin states for given m -value. When all these 2(2 l 1) states are filled, we speak of the closing of a shell. The charge density for a closed shell has the form
(14A-13)
and this is spherically symmetric because of the property of spherical harmonics that
(14A-14)
l m l
Ylm ( , )^2 2 l 1 4
e
l m l
Rnl ( r )^2 Ylm ( , )^2
W-54 Supplement 14-A The Hartree Approximation
Supplement 14-B
The Building-Up Principle
In this section we discuss the building up of atoms by the addition of more and more elec- trons to the appropriate nucleus, whose only role, to good approximation, is to provide the positive charge Ze.
Hydrogen (Z 1) There is only one electron, and the ground-state configuration is (1 s ). The ionization energy is 13.6 eV, and the amount of energy needed to excite the first state above the ground state is 10.2 eV. The radius of the atom is 0.5 Å, and its spectroscopic description is 2 S 1/.
Helium (Z 2) The lowest two-electron state, as we saw in Chapter 14, is one in which both electrons are in the (1 s ) orbital. We denote this configuration by (1 s )^2. In spectro- scopic notation, the ground state is an l 0 spin singlet state, 1 S 0 because the exchange effect favors it. The total binding energy is 79 eV. After one electron is removed, the re- maining electron is in a (1 s ) orbit about a Z 2 nucleus. Thus its binding energy is 13.6 Z^2 eV 54.4 eV, and the energy required to remove the first electron, the ionization energy , is 79.0 54.4 24.6 eV. A rough estimate of the energy of the first excited state, with configuration (1 s )(2 s ), is 13.6 Z^2 13.6( Z 1)^2 / n^2 58 eV for Z 2 and n 2. This expression takes into account shielding in the second term. Thus the excitation en- ergy is 79 eV 58 eV 21 eV.^1 In any reaction with another substance, about 20 eV is required for a rearrangement of the electrons, and thus helium is chemically very inactive. This property is shared by all atoms whose electrons form closed shells, but the energy re- quired is particularly large for helium.
Lithium (Z 3) The exclusion principle forbids a (1 s ) 3 configuration, and the lowest energy electron configuration is (1 s ) 2 (2 s ). We are thus adding an electron to a closed shell, and since the shell is in a 1 S 0 state, the spectroscopic description of the ground state is 2 S 1/2 , just as for hydrogen. If the screening were perfect, we should expect a binding energy of 3.4 eV (since n 2). The screening is not perfect, especially since the outer valence electron being in an s -state, its wave function has a reasonable overlap with nu- cleus at r 0. We can estimate the effective Z from the measured ionization energy of 5.4 eV, and it is Z * 1.3. It takes very little energy to excite the lithium atom. The six (2 p ) electronic states lie just a little above the (2 s ) state, and these (2 p ) states, when oc- cupied, make the atom chemically active (see our more extended discussion of carbon). Lithium, like other elements that have one electron outside a closed shell, is a very active element.
W-
(^1) This is a crude estimate that ignores the electron–electron repulsion and exchange effects. The difference between the 21 eV and the 24.6 eV is the 4–5 eV that will be released when the excited atom decays to its ground state. (See Fig. 14-2 b .)
Nitrogen (Z 7) Here the configuration is (1 s )^2 (2 s )^2 (2 p )^3 , sometimes described as (2 p )^3 for brevity (the closed shells and subshells are omitted). The three electrons can all be in nonoverlapping p -states, and thus we expect the increase in ionization energy to be the same as the increase from boron to carbon. This is in agreement with the measured value of 14.5 eV.
Oxygen (Z 8) The configuration may be abbreviated to (2 p )^4 , and the shell is more than half full. Since there are four electrons, it appears as if the determination of the ground-state spectroscopic state would be very difficult. We can, however, look at the shell in another way. We know that when two more electrons are added to make a (2 p )^6 configuration, then the shell is filled, and the total state has L S 0. We can thus think of oxygen as having a closed 2 p shell with two holes in it. These holes are just like anti- electrons, and we can look at two-hole configurations. These will be the same as two- electron configurations, since the holes also have spin 1/2. Thus, as with carbon, the possible states consistent with the antisymmetry of the two-fermion (two-hole) wave function are 1 S , 3 P , 1 D , and the four electrons must be in the same states, since they, to- gether with the two-hole system, give S 0, L 0. The highest spin is S 1, and thus we must have a 3 P state. Hund’s rule, which will be discussed in the next section, yields the 3 P 2 state. When the fourth electron is added to the nitrogen configuration, it must go into an orbital with an m -value already occupied. Thus two of the electron wave functions overlap, and this raises the energy because of the repulsion. It is therefore not surprising that the ionization energy drops to the value of 13.6 eV.
Fluorine (Z 9) Here the configuration is (2 p )^5. The monotonic increase in the ioniza- tion energy resumes, with the experimental value of 17.4 eV. Fluorine is chemically very active, because it can “accept” an electron to form a closed shell (2 p )^6 , which is very sta- ble. Since the addition of a single electron with s 1/2 and l 1 yields a 1 S 0 state, the shell with the hole in it must have s 1/2 and l 1. It is therefore a 2 P state, and by Hund’s rule, as we shall see, the state is 2 P 3/.
Neon (Z 10) With Z 10 the (2 p ) shell is closed, and all electrons are paired off. The ionization energy is 21.6 eV, continuing the monotonic trend. Here, as in helium, the first available state that an electron can be excited into has a higher n value, and thus it takes quite a lot of energy to perturb the atom. Neon, like helium, is an inert gas. At this point, the addition of another electron requires putting it in an orbit with a higher n value ( n 3), and thus neon marks the end of a period in the periodic table, as did helium. In neon, as in helium, the first available state into which an electron can be ex- cited has a higher n -value, so that it takes quite a lot of energy to perturb the atom. Neon shares with helium the property of being an inert gas.
The Building-Up Principle W-
z
y
x
cos 2
sin 2 θcos 2
θ
φ sin 2 θ sin 2 φ Figure 14B-1 Orthogonal distributions of electronic clouds in the l 1 state.
The next period again has eight elements in it. First the (3 s ) shell is filled, with sodium ( Z 11) and magnesium ( Z 12), and then the 3 p shell, which includes, in order, aluminum ( Z 13), silicon ( Z 14), phosphorus ( Z 15), sulphur ( Z 16), chlorine ( Z 17) and, closing the shell, argon ( Z 18). These elements are chemi- cally very much like the series: lithium,... , neon, and the spectroscopic descriptions of the ground states are the same. The only difference is that, since n 3, the ioniza- tion energies are somewhat smaller, as can be seen from the periodic table at the end of this supplement. It might appear a little strange that the period ends with argon, since the (3 d ) shell, accommodating ten elements, remains to be filled. The fact is that the self- consistent potential is not of the 1/ r form, and the intrashell splitting here is suffi- ciently large that the (4 s ) state lies lower than the (3 d ) state, though not by much. Hence a competition develops, and in the next period we have (4 s ), (4 s ) 2 , (4 s ) 2 (3 d ), (4 s ) 2 (3 d ) 2 , (4 s ) 2 (3 d ) 3 , (4 s )(3 d ) 5 , (4 s ) 2 (3 d ) 5 , (4 s ) 2 (3 d ) 6 , (4 s ) 2 (3 d ) 7 , (4 s ) 2 (3 d ) 8 , (4 s )(3 d ) 10 , (4 s ) 2 (3 d ) 10 and then the 4 p shell gets filled until the period ends with krypton ( Z 36). The chemical properties of elements at the beginning and end of this period are similar to those of elements at the beginning and end of other periods. Thus potassium, with the single (4 s ) electron, is an alkali metal, like sodium with its single (3 s ) electron out- side a closed shell. Bromine, with the configuration (4 s ) 2 (3 d ) 10 (4 p ) 5 , has a single hole in a p -shell and thus is chemically like chlorine and fluorine. The series of elements in which the (3 d ) states are being filled all have rather similar chemical properties. The reason for this again has to do with the details of the self-consistent potential. It turns out that the radii of these orbits 2 are somewhat smaller than those of the (4 s ) electrons, so that when the (4 s ) 2 shell is filled, these electrons tend to shield the (3 d ) electrons, no matter how many there are, from outside influences. The same effect occurs when the (4 f ) shell is being filled, just after the (6 s ) shell has been filled. The elements here are called the rare earths.
Spectroscopic Description of Ground States
In our discussion of the light atoms, we often gave the spectroscopic description of the ground states—for example, 3 P 2 for oxygen, 2 P 3/2 for fluorine, and so on. The knowledge of S , L , and J for the ground states is important, because selection rules allow us to deter- mine these quantities for the excited states of atoms. We referred to Hund’s rules in the determination of these, and these rules are the subject of this section. What determines the ground-state quantum numbers is an interplay of spin-orbit coupling and the exchange effect discussed in connection with helium in Chapter 14. For the lighter atoms ( Z 40), for which the motion of the electrons is nonrelativistic, the electron–electron repulsion effects are more important than the spin-orbit coupling. This means that it is a fairly good approximation to view L and S as separately good quantum numbers: We add up all the spins to form an S and all the orbital angular mo- menta of the electrons to form an L , and these are then coupled to obtain a total J. For heavier atoms it is a better approximation to first couple the spin and orbital angular momentum to form a total angular momentum for that electron, and then to couple all of the J ’s together. The former case is described as Russell-Saunders coupling, the lat- ter as j - j coupling. For Russell-Saunders coupling, F. Hund summarized the results of
W-58 Supplement 14-B The Building-Up Principle
(^2) It is understood that this is just a way of talking about the peaking tendencies of the charge distribution.
For manganese the shelves have Lz 2, 1, 0, 1, 2, as shown in Fig. 14B-3. There are five electrons, and thus each of the spaces is filled by one. With the spins parallel we get Sz 5/2, which implies that S 5/2. The total value of Lz 0, and thus we have an S -state. This means that the ground state is an 6 S 5/2 state. Limitations of space prevent us from a more detailed discussion of the periodic table. A few additional comments are, however, in order. (a) There is nothing in atomic structure that limits the number of elements. The rea- son that atoms with Z 100 do not occur naturally is that heavy nuclei undergo spontaneous fission. If new, superheavy (metastable) nuclei are ever discovered, there will presumably exist corresponding atoms, and it is expected that their structure will conform to the prediction of the building-up approach outlined in this supplement. (b) The ionization energies all lie in the vicinity of 5–15 eV. The reason is that in spite of the increasing number of electrons, the outermost electrons “see” a charge that lies in the range Z 1 – 2. In addition, because of the departures from a point charge distribution, the dependence of the energy is no longer of the 1/ n^2 form. Consequently, the wave functions of the outermost electrons do not extend much further than that of the electron in the hydrogen atom. Atoms are more or less the same size! (c) We went to a great deal of trouble to specify the S , L , and J quantum numbers of the ground states of the various elements. The reason for doing this is that in spectroscopy, the quantum numbers are of particular interest because of the se- lection rules
(14B-2)
that will be derived later, and that may then be used to determine the quantum numbers of the excited states. The spectroscopy of atoms, once we get beyond hydrogen and helium, is very complicated. Consider, as a relatively simple ex- ample, the first few states of carbon, which are formed from different configura- tions of the two electrons that lie outside the closed shell in the (2 p )^2 orbitals. As already pointed out, the possible states are 1 S 0 , 3 P 2,1,0, and 1 D 2. The 3 P 0 state lies lowest, but the other states are still there. The first excited states may be de- scribed by the orbitals (2 p )(3 s ). Here S 0 or 1, but L 1 only. Since the n -values are different, the exclusion principle does not restrict the states in any way, and all of the states 1 P 1 , 3 P 2,1,0 are possible, while the excited states that
J 0, 1 (no 0 0)
L 1
S 0
W-60 Supplement 14-B The Building-Up Principle
2
1
Mn 0
- 1
- 2 Figure 14B-3^ Application of Hund’s rules to atom with^ five valence electrons in d -state.
arise from the orbitals (2 p )(3 p ) can have S 0, 1 and L 2, 1, 0, leading to all the states 1 D 2 , 1 P 1 , 1 S 0 , 3 D 3,2,1 , 3 P 2,1,0, and 3 S 1. Even with the restrictions provided by the selection rules, there are numerous transitions. Needless to say, the order- ing of these levels represents a delicate balance between various competing ef- fects, and the prediction of the more complex spectra is very difficult. That task is not really of interest to us, since the main point that we want to make is that quantum mechanics provides a qualitative, and quantitative, detailed explanation of the chemical properties of atoms and of their spectra, without assuming an in- teraction other than the electromagnetic interaction between charged particles. We shall have occasion to return to the topic of spectra.
Periodic Table
Ionization Radius 2 Z Element Configuration Term^1 Potential eV in Å
1 H (1 s ) 2 S 1/2 13.6 0. 2 He (1 s )^2 1 S 0 24.6 0. 3 Li (He)(2 s ) 2 S 1/2 5.4 1. 4 Be (He)(2s) 2 1 S 0 9.3 1. 5 B (He)(2 s )^2 (2 p ) 2 P 1/2 8.3 0. 6 C (He)(2 s )^2 (2 p )^2 3 P 0 11.3 0. 7 N (He)(2 s )^2 (2 p )^3 4 S 3/2 14.5 0. 8 O (He)(2 s )^2 (2 p )^4 3 P 2 13.6 0. 9 F (He)(2 s )^2 (2 p )^5 2 P 3/2 17.4 0. 10 Ne (He)(2 s )^2 (2 p )^6 1 S 0 21.6 0. 11 Na (Ne)(3 s ) 2 S 1/2 5.1 1. 12 Mg (Ne)(3 s )^2 1 S 0 7.6 1. 13 Al (Ne)(3 s )^2 (3 p ) 2 P 1/2 6.0 1. 14 Si (Ne)(3 s )^2 (3 p )^2 3 P 0 8.1 1. 15 P (Ne)(3 s )^2 (3 p )^3 4 S 3/2 11.0 0. 16 S (Ne)(3 s )^2 (3 p )^4 3 P 2 10.4 0. 17 Cl (Ne)(3 s )^2 (3 p )^5 2 P 3/2 13.0 0. 18 Ar (Ne)(3 s )^2 (3 p )^6 1 S 0 15.8 0. 19 K (Ar)(4 s ) 2 S 1/2 4.3 2. 20 Ca (Ar)(4 s )^2 1 S 0 6.1 1. 21 Sc (AR)(4 s )^2 (3 d ) 2 D 3/2 6.5 1. 22 Ti (Ar)(4 s )^2 (3 d )^2 3 F 2 6.8 1. 23 V (Ar)(4 s )^2 (3 d )^3 4 F 3/2 6.7 1. 24 Cr (Ar)(4 s )(3 d )^5 7 S 3 6.7 1. 25 Mn (Ar)(4 s )^2 (3 d )^5 6 S 5/2 7.4 1. 26 Fe (Ar)(4 s )^2 (3 d )^6 5 D 4 7.9 1. 27 Co (Ar)(4 s )^2 (3 d )^7 4 F 9/2 7.8 1. 28 Ni (Ar)(4 s )^2 (3 d )^8 3 F 4 7.6 1. 29 Cu (Ar)(4 s )(3 d )^10 2 S 1/2 7.7 1. 30 Zn (Ar)(4 s )^2 (3 d )^10 1 S 0 9.4 1. 31 Ga (Ar)(4 s )^2 (3 d )^10 (4 p ) 2 P 1/2 6.0 1.
( Continued )
The Building-Up Principle W-
(^1) Term designation is equivalent to spectroscopic description.
- Ionization Radius
- 32 Ge (Ar)(4 s )^2 (3 d )^10 (4 p )^2 3 P 0 8.1 1. Z Element Configuration Term^1 Potential eV in Å
- 33 As (Ar)(4 s )^2 (3 d )^10 (4 p )^3 4 S 3/2 10.0 1.
- 34 Se (Ar)(4 s )^2 (3 d )^10 (4 p )^4 3 P 2 9.8 0.
- 35 Br (Ar)(4 s )^2 (3 d )^10 (4 p )^5 2 P 3/2 11.8 0.
- 36 Kr (Ar)(4 s )^2 (3 d )^10 (4 p )^6 1 S 0 14.0 0.
- 37 Rb (Kr)(5 s ) 2 S 1/2 4.2 2.
- 38 Sr (Kr)(5 s )^2 1 S 0 5.7 1.
- 39 Y (Kr)(5 s )^2 (4 d ) 2 D 3/2 6.6 1.
- 40 Zr (Kr)(5 s )^2 (4 d )^2 3 F 2 7.0 1.
- 41 Nb (Kr)(5 s )(4 d )^4 6 D 1/2 6.8 1.
- 42 Mo (Kr)(5 s )(4 d )^5 7 S 3 7.2 1.
- 43 Tc (Kr)(5 s )^2 (4 d )^5 6 S 5/2 Not known 1.
- 44 Ru (Kr)(5 s )(4 d )^7 5 F 5 7.5 1.
- 45 Rh (Kr)(5 s )(4 d )^8 4 F 9/2 7.7 1.
- 46 Pd (Kr)(4 d )^10 1 S 0 8.3 0.
- 47 Ag (Kr)(5 s )(4 d )^10 2 S 1/2 7.6 1.
- 48 Cd (Kr)(5 s )^2 (4 d )^10 1 S 0 9.0 1.
- 49 In (Kr)(5 s )^2 (4 d )^10 (5 p ) 2 P 1/2 5.8 1.
- 50 Sn (Kr)(5 s )^2 (4 d )^10 (5 p )^2 3 P 0 7.3 1.
- 51 Sb (Kr)(5 s )^2 (4 d )^10 (5 p )^3 4 S 3/2 8.6 1.
- 52 Te (Kr)(5 s )^2 (4 d )^10 (5 p )^4 3 P 2 9.0 1.
- 53 I (Kr)(5 s )^2 (4 d )^10 (5 p )^5 2 P 3/2 10.4 1.
- 54 Xe (Kr)(5 s )^2 (4 d )^10 (5 p )^6 1 S 0 12.1 0.
- 55 Cs (Xe)(6 s ) 2 S 1/2 3.9 2.
- 56 Ba (Xe)(6 s )^2 1 S 0 5.2 2.
- 57 La (Xe)(6 s )^2 (5 d ) 2 D 3/2 5.6 1.
- 58 Ce (Xe)(6 s )^2 (4 f )(5 d ) 3 H 5 6.9 1.
- 59 Pr (Xe)(6 s )^2 (4 f )^3 4 I 9/2 5.8 1.
- 60 Nd (Xe)(6 s )^2 (4 f )^4 5 I 4 6.3 1.
- 61 Pm (Xe)(6 s )^2 (4 f )^5 6 H 5/2 Not known 1.
- 62 Sm (Xe)(6 s )^2 (4 f )^6 7 F 0 5.6 1.
- 63 Eu (Xe)(6 s )^2 (4 f )^7 8 S 7/2 5.7 1.
- 64 Gd (Xe)(6 s )^2 (4 f )^7 (5 d ) 9 D 2 6.2 1.
- 65 Tb (Xe)(6 s )^2 (4 f )^9 6 H 15/2 6.7 1.
- 66 Dy (Xe)(6 s )^2 (4 f )^10 5 I 8 6.8 1.
- 67 He (Xe)(6 s )^2 (4 f )^11 4 I 15/2 Not known 1.
- 68 Er (Xe)(6 s )^2 (4 f )^12 3 H 6 Not known 1.
- 69 Tm (Xe)(6 s )^2 (4 f )^13 2 F 7/2 Not known 1.
- 70 Yb (Xe)(6 s )^2 (4 f )^14 1 S 0 6.2 1.
- 71 Lu (Xe)(6 s )^2 (4 f )^14 (5 d ) 2 D 3/2 5.0 1.
- 72 Hf (Xe)(6 s )^2 (4 f )^14 (5 d )^2 3 F 2 5.5 1.
- 73 Ta (Xe)(6 s )^2 (4 f )^14 (5 d )^3 4 F 3/2 7.9 1.
- 74 W (Xe)(6 s )^2 (4 f )^14 (5 d )^4 5 D 0 8.0 1.
- 75 Re (Xe)(6 s )^2 (4 f )^14 (5 d )^5 6 S 5/2 7.9 1.
- 76 Os (Xe)(6 s )^2 (4 f )^14 (5 d )^6 5 D 4 8.7 1. - Ionization Radius Periodic Table
- 77 Ir (Xe)(6 s )^2 (4 f )^14 (5 d )^7 4 F 9/2 9.2 1. Z Element Configuration Term^1 Potential eV in Å
- 78 Pt (Xe)(6 s )(4 f )^14 (5 d )^9 3 D 3 9.0 1.
- 79 Au (Xe)(6 s )(4 f )^14 (5 d )^10 2 S 1/2 9.2 1.
- 80 Hg (Xe)(6 s )^2 (4 f )^14 (5 d )^10 1 S 0 10.4 1.
- 81 Tl (Xe)(6 s )^2 (4 f )^14 (5 d )^10 (6 p ) 2 P 1/2 6.1 1.
- 82 Pb (Xe)(6 s )^2 (4 f )^14 (5 d )^10 (6 p )^2 3 P 0 7.4 1.
- 83 Bi (Xe)(6 s )^2 (4 f )^14 (5 d )^10 (6 p )^3 4 S 3/2 7.3 1.
- 84 Po (Xe)(6 s )^2 (4 f )^14 (5 d )^10 (6 p )^4 3 P 2 8.4 1.
- 85 At (Xe)(6 s )^2 (4 f )^14 (5 d )^10 (6 p )^5 2 P 3/2 Not known 1.
- 86 Rn (Xe)(6 s )^2 (4 f )^14 (5 d )^10 (6 p )^6 1 S 0 10.7 1.
- 87 Fr (Rn)(7 s ) Not known 2.
- 88 Ra (Rn)(7 s )^2 1 S 0 5.3 2.
- 89 Ac (Rn)(7 s )^2 (6 d ) 2 D 3/2 6.9 1.
- 90 Th (Rn)(7 s )^2 (6 d )^2 3 F
- 91 Pa (Rn)(7 s )^2 (5 f )^2 (6 d ) 4 K 11/
- 92 U (Rn)(7 s )^2 (5 f )^3 (6 d ) 5 L
- 93 Np (Rn)(7 s )^2 (5 f )^4 (6 d ) 6 L 11/
- 94 Pu (Rn)(7 s )^2 (5 f )^6 7 F
- 95 Am (Rn)(7 s )^2 (5 f )^7 8 S 7/
- 96 Cm (Rn)(7 s )^2 (5 f )^7 (6 d ) 9 D
- 97 Bk (Rn)(7 s )^2 (5 f )^9 6 H 15/
- 98 Cf (Rn)(7 s )^2 (5 f )^10 5 I
- 99 Es (Rn)(7 s )^2 (5 f )^11 4 I 15/
- 100 Fm (Rn)(7 s )^2 (5 f )^12 3 H
- 101 Md (Rn)(7 s )^2 (5 f )^13 2 F 7/
- 102 No (Rn)(7 s )^2 (5 f )^14 1 S - The Building-Up Principle W- 2 Radius is defined by the peak of the calculated charge density of the outermost orbital.
Supplement 14-C
A Brief Discussion of Molecules
The purpose of this supplement is to outline the basic approach to the study of simple molecules. We discuss the H 2 molecule in some detail, so as to provide an understanding of terms like molecular orbitals and valence bonds. Quantum chemistry has become a field well served by massive computers. Our discussion does not really provide an entry into this field. It is extremely simple-minded, and its only justification is that it provides an insight into the basic mechanisms that lead to molecular binding. Anything more de- pends on an understanding of electron–electron correlations, and these are way beyond the scope of this book. Our approach will follow the one followed in the discussion of the molecule. It is based on the fact that the nuclei are much more massive than the elec- trons, and that therefore a good first approximation treats the nuclei as frozen , with their location determined by the electronic distribution. We discuss the H 2 molecule as a proto- type of other simple diatomic molecules.
The H 2 Molecule
The H 2 molecule is a more complicated system, because there are two electrons present, and the exclusion principle therefore plays a role. As in the case of the molecule, we treat the nuclei as fixed. The nuclei (protons here) will be labeled A and B , and the two electrons 1 and 2, re- spectively (Fig. 14C-1). The Hamiltonian has the form
(14C-1)
where
(14C-2)
depends only on the coordinates of the electron i relative to the nuclei. We will again compute an upper bound to E ( RAB ) by constructing the expectation value of H with a trial wave function. Since
(14C-3)
are just Hamiltonians for the molecule (14-50) it seems reasonable to take as our trial wave function a product of two functions of the type shown in the first line of (14-51):
g ( r 1 , r 2 ) 1 (14C-4) 2[1 S ( RAB )] [ A ( r 1 ) B ( r 1 )][ A ( r 2 ) B ( r 2 )] X singlet
H 2
H^ ˜^ i Hi e
2 4 0 RAB
Hi
p^2 i 2 m e
2 4 0 rAi e
2 4 0 rBi ( i 1, 2)
H H 1 H 2 e
2 4 0 r 12 e
2 4 0 RAB
H 2
H 2
W-
The Valence Bond Method
The last difficulty can be avoided with the use of the valence bond (also called Heitler- London) method, in which linear combinations of atomic orbitals are used. The singlet wave function used as a trial wave function in the variational principle is taken to be
(14C-7)
where, as before, the A ( r i ) are hydrogenic wave functions for the i th electron about pro- ton A. We could, in principle, add a triplet term to our variational trial wave function. However, a triplet wave function must be spatially antisymmetric and yields low proba- bility for the electrons being located in the region between the protons. We saw in our dis- cussion of the molecule that just this configuration led to the lowest energy. Although it is not immediately obvious that the attraction is still largest in this configuration when there are two electrons that repel each other in the system, it is in fact so. The results of a variational calculation with the VB trial wave function are
This is not a significant improvement over the MO results, for the simple reason that the inadequacy of the trial wave functions for small RAB carries more weight. There should be no question about the quantitative successes of quantum mechanics in molecular physics. More sophisticated trial wave functions have to be used; for example, a 50-term trial wave function yields complete agreement with observations for the H 2 molecule, but it does not, as the MO and VB functions do, give us something of a qualitative feeling of what goes on between the atoms. In what follows, we will explore the relevance of these approaches to a qualitative understanding of some aspects of chemistry. The expectation value of H for the H 2 molecule in the VB approach has the following schematic form:
where Ti is the kinetic energy of the i th electron, and since
(^) A 1 B 2 2 E 1 e
2 4 0 rB 2 e
2 4 0 rA 1 e
2 4 0 r 12 e
2 4 0 RAB ^ A 2 B 1
1 S^2
A 1 B 2 ^2 E 1 ^ e^2 4 0 rB 1 e
2 4 0 rA 2 e
2 4 0 r 12 e
2 4 0 RAB ^ A 1 B 2
H 1
2(1 S^2 )
A 1 B 2 A 2 B 1 H A 1 B 2 A 2 B 1
T^1 ^
e^2 4 0 rA 1
A 1 E 1 A 1
e
2 4 0 r 12
e
2 4 0 RAB ^ A 1 B 2 ^ A 2 B 1
1 S^2
A 1 B 2 T 1 ^ T 2 ^ e^2 4 0 rA 1 e
2 4 0 rA 2 e
2 4 0 rB 1 e
2 4 0 rB 2
H 1
2(1 S^2 )
A 1 B 2 A 2 B 1 H A 1 B 2 A 2 B 1
R 0.87 Å
Eb 3.14 eV
H 2
( r 1 , r 2 ) 1 2[1 S^2 ( RAB )]
1/ [ A ( r 1 ) B ( r 2 ) A ( r 2 ) B ( r 1 )] X singlet
W-66 Supplement 14-C A Brief Discussion of Molecules
(14C-8)
In obtaining this, liberal use has been made of symmetry. The terms that can make this expression more negative are
The former is just the attraction of the electron cloud about one proton to the other proton; the second is the overlap of the two electrons (weighted with 1/ rA 1 ). If this can be large, there will be binding. The two electrons can only overlap significantly, however, if their spins are antiparallel; this is a consequence of the exclusion principle. The region of over- lap is between the two nuclei, and there the attraction to the nuclei generally overcomes the electrostatic repulsion between the electrons. In the MO picture, too, it is an overlap term—the result in (14-60)—that is crucial to bonding, and again, bonding occurs because the electron charge distribution is large be- tween the nuclei. Thus, although here the orbitals belong to the whole molecule rather than to individual atoms, the physical reason for bonding is the same. We note that in general there may be several bound states of the nuclei, correspond- ing to different electronic configurations. For example, if in (14C-7) we take the ( r 2 ) wave function to be a u 200 eigenfunction, while the ( r 1 ) remains a u 100 eigenfunction, the overlap may be such as to provide a second, more weakly bound state of the protons. We are not going to pursue this, except to point out the important fact that the E ( R ) is different for each electronic state.
The Importance of Unpaired Valence Electrons
An important simplification in the study of electronic charge distributions in molecules occurs because we really do not need to take all electrons into account. In the construction of orbitals, be it valence or molecular, only the outermost electrons, not in closed shells— that is, the so-called valence electrons —have a chance to contribute to the bonding. The inner electrons, being closer to the nucleus, are less affected by the presence of another atom in the vicinity. 1 Furthermore, not all valence electrons contribute equally: if two electrons are in a spin 0 state—we call them paired electrons —they will not give rise to bonding. To see why this is so, consider what happens when an atom with a single va- lence electron is brought near an atom with two paired electrons. There are two cases to be considered (Fig. 14C-2). (a) If the two electrons that are parallel exchange [i.e., are put into a form such as (14C-7) with a minus sign between the terms], then they must be in a triplet
A 1
rB 1 A^1 ^ and^
S
1 S^2
A 1
rA 1 B^1
* A 1 B 1 * B 2 A 2
r 12
2 e
2 4 0 S A 1 (^) rA^11 B 1 e
2 4 0
A 1 ^2 B 2 ^2
r 12
1 S^2
2 E 1 e
2 4 0 rRAB (1 S^2 ) 2 e
2 4 0 A 1 (^) r^1 B 1 A 1
A Brief Discussion of Molecules W-
(^1) It may happen in atoms that even the valence electrons are rather close to the nucleus. This is the case for the rare earths. A consequence of the fact that the outer electrons in 5 d and 4 f shells lie close in is that the rare earths are chemically less active than the transition metals ( Z 20 – 30).
