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7.1 Electronic states of helium atom
Neutral He atom ( Z =2) has 2 electrons. The electronic Hamiltonian in a.u. is (Figure 7.1)
̂ eq 7.1a
r (^2) r 1
r 12
O
e 1
e 2 ( x 1 , y 1 , z 1 )
( x 2 , y 2 , z 2 )
Figure 7.1 Interparticle distances in the helium atom. The nucleus is at the origin of a Cartesian coordinate system (axes not shown). The nucleus is fixed in space; only the electrons are moving.
The last term in eq 7.1a is the contribution to the potential energy arising from the repulsion between the two electrons; the value of this term depends on the coordinates of both electrons whereas the first four terms depend on coordinates of individual electrons only. We rewrite eq 7.1a in the form
̂̂ ( )̂ ( ) eq 7.1b
where
̂( ) eq 7.
depends on the coordinates of the jth^ electron only. The operator ̂ is recognized as the Hamiltonian of an electron in a hydrogenlike atom (with Z =2, in this case; see eqs 6.41-43). The two operators ̂( ) (^) and ̂ ( ) (^) trivially commute with each other because they depend on different variables.
However, neither one commutes with the interaction term 1/r 12. Because of this latter fact, the Schrödinger equation for the helium is not separable, and we must use approximation methods.
7.2 The Variation Method
The “variation method” is a powerful procedure that forms the basis of most quantum-mechanical calculations made on atoms and molecules. It is based on the “variation principle”.
Let ̂ be a time-independent Hamiltonian operator, and E 1 be the exact lowest eigenvalue of ̂ (i.e. the exact ground state energy). Given a normalized wavefunction, , we can calculate an energy by the average value expression
∫ ̂ normalized eq 7.3a
Note that in this expression, ̂ is the exact Hamiltonian whereas is an approximate function that we design (i.e. we guess it). The variation principle states that can never be less than the true lowest energy:
Variation Principle eq 7.3b
The normalized function is called a “trial variation function”, and the integral in eq 7.3a is called the “variational integral”. In order to achieve a good approximation to the ground state energy, we try many different trial functions and look for the one that gives the lowest value for the variational integral. The approximate functions that we devise for this purpose must have certain characteristic properties that are known about the exact solutions of the Schrödinger equation.
7.3 Total Orbital and Spin Angular Momenta in an Atom
We can obtain very useful information about the features of the exact wavefunctions if there are available other operators that commute with the Hamiltonian of the system. This is because when̂ commutes with another operator ̂, the two operators have simultaneous eigenfunctions. It is generally the case that the eigenvalues and the eigenfunctions of ̂ can be found much more easily than solving the Schrödinger equation, ̂. The fact that must also be an eigenfunction of ̂ means that can be characterized by the eigenvalues (and eigenfunctions) of ̂. It is thus
important to search for operators that commute with ̂. We now discuss properties of several operators of this type.
7.3.1 The total orbital angular momentum of electrons in an atom
The total orbital angular momentum of N electrons in an atom is a vector operator defined by
̂⃗̂⃗ ̂⃗ ̂⃗ eq 7.
where ̂⃗ is the orbital angular momentum vector operator of the jth^ electron. The z-component of̂⃗ is
̂ ̂ ̂ ̂ eq 7.
As in the case for a single electron, it may be shown that ̂ commutes with ̂ , and that both of these operators also commute with the Hamiltonian ̂ of the atom. It follows that the exact solutions, , of the Schrödinger equation are eigenfunctions of ̂ and ̂. It is found that eigenvalues of the latter operators are (in a.u.) as in Table 7.1:
Table 7.1 Eigenvalues of orbital angular momentum operators (a.u.)
Operator ̂ ̂
Eigenvalues L ( L +1) where L =0, 1, 2,… ML where – L ML L
Thus the exact atomic functions can be classified according to the L and ML values they possess. It turns out that the total electronic energy, E , of a many-electron atom is independent of the ML quantum number, but it does depend on L (for N >1). The spatial degeneracy of an atomic energy
For a single electron ( N =1), eigenvalues in Table 7.2 reduce to those given previously. Note the usage of lowercase and uppercase symbols for the quantum numbers, depending on whether we have a single electron or many electrons. This is a general convention.
The Hamiltonian operator of the atom that we are considering, e.g. eq 7.1 for He, is a function of only the spatial variables of the electrons whereas ̂ and ̂ are functions of the spin variables. Therefore, the latter operators trivially commute with ̂. Hence the state functions of an atom must be eigenfunctions of ̂ and ̂ ; and as a result, can be labeled by the spin quantum numbers S and MS , in addition to L and ML.
Similar to the case with the orbital angular momentum, the total electronic energy of an atom does not depend on the MS quantum number, but it does depend on the total spin quantum number S (for N >1). There are 2 S +1 different values of MS for a given S. Thus the “spin degeneracy” of an atomic energy level E is 2 S +1 with the total degeneracy being (2 L +1)(2 S +1). The spin degeneracy, 2 S +1, is also called the “spin multiplicity” or simply, the multiplicity of the atomic state.
As an example let us consider the allowed electronic states in the helium atom. Here, N =2 so that there are only two values for S : 0 and 1. For S =0, the spin multiplicity is 1 (“singlet”), and for S =1 the multiplicity is 3 (“triplet”). Thus the complete set of states can be grouped into two types as distinguished by their multiplicities: a group of “singlet states”, and another group consisting of “triplet states”. These states are also characterized by an L quantum number associated with each one of them. In contrast to S which takes only two values in the He atom, L may take infinitely many different values: L =0, 1, 2,… Angular momentum characteristics of a state are shown by combining the multiplicity and L in the following form:
2 S +1 L Atomic Term Symbol eq 7.
where a letter code is used for L. The letters employed are the uppercases of those used for the orbitals. Thus, in the He atom, the term symbols of the singlet states are 1 S, 1 P, …, and those of the triplets are 3 S, 3 P,… Note that there are infinitely many states of each of these term types; i.e. many (^1) S states with different energies, many 3 S states, etc.
Exercise 7.1 List all possibilities of the spin multiplicities for each of the atoms: a) Li, b) Be, c) N.
7.3.3 The Pauli Exclusion Principle
In Section 3.3 we stated this principle in terms of the maximum occupation number of an orbital energy level as nmax =2 g where g is the degeneracy of the orbital energy. Since by definition, g is the number of different orbitals having the same energy, this form of the Pauli principle means that the maximum occupation number of a single orbital is 2. Here we will give a more general definition of this principle.
Let us consider the two electrons in the He atom. We label the electrons by the integers 1 and 2, and write the wavefunction as (1,2) where the arguments stand for all four independent variables ( position+1 spin variable) for each electron. The general statement of the Pauli principle is:
The wavefunction (1,2) must change sign when the labels of the two electrons are exchanged,
(2,1) = (1,2) Pauli exclusion principle eq 7.
For the purpose of gaining a better insight about the meaning of eq 7.10, let us introduce the permutation operator ̂ that exchanges the labels of the two electrons in any function f (1,2) of their variables; i.e. ̂ ( ) ( ). Since applying ̂ twice in a row to f (1,2) restores the function to its original value, we have ̂ , i.e. the identity operator. Finding eigenvalues of ̂ is easy. Assume that f is an eigenfunction of ̂ with eigenvalue p :
̂
Applying ̂ to both sides of this equation, we have
̂ ( ̂ )
Since the squared operator on the left is the identity operator, one has f = p^2 f , or p =±1. The eigenfunctions with p =+1 do not change sign in the exchange of the two electrons; they are said to be “symmetric” with respect to the exchange. Those eigenfunctions with p =-1 change sign in the interchange, and are said to be “antisymmetric” with respect to the exchange. We see from eq 7. that (1,2) must be an eigenfunction of ̂ with eigenvalue p =-1; i.e. it must be antisymmetric when all four variables of the two electrons are exchanged.
You should easily see that if you exchange the electron labels in eq 7.1a or b, you will obtain the same expression. Because of this property, the Hamiltonian operator of an atom commutes with the permutation operator. Note that ̂ also commutes with the operators ̂ ̂ , ̂ and ̂. It follows that ̂ and ̂ have simultaneous eigenfunctions; i.e. the state functions (1,2) can be labeled by p. According to the Pauli principle, only those (1,2) with p =-1 are physically acceptable.
7.4 Designing trial variation functions for the helium atom
The trial functions that are used in the variational integral (eq 7.3a) must have the properties of exact wavefunctions outlined above: a) trial must be antisymmetric with respect to the exchange of the variables of the two electrons; b) trial must be a simultaneous eigenfunction of the four operators: ̂ ̂ , ̂ and ̂. We discuss first the antisymmetry requirement.
The exact wavefunction for a two-electron system such as the He atom can be written as the product of a space-dependent part and a spin-dependent part:
(1,2) = space (1,2) spin (1,2) eq 7.
where space (1,2) is a function of only the spatial variables of the two electrons, and spin (1,2) is a function of only the spin variables. For the total wavefunction (1,2) to obey the Pauli principle, one
are two functions, αβ and βα, with MS =0 in the table. However, they are not eigenfunctions of ̂ as required by spin (1,2) of Table 7.3. When there are two or more independent functions at some stage of a problem, the actual expression for the “true” function is obtained by using extra information known about it. In this case, we know that they have to be eigenfunctions of ̂ with an eigenvalue +1 or -1. To find them, we form the linear combination of αβ and βα
= c 1 (1)(2) + c 2 (1)(2)
where the coefficients c 1 and c 2 are to be (partially) determined by requiring that ̂ = p with p =±1. You should verify that for p =1, c 2 =c 1 , whereas for p =-1, c 2 =-c 1. We thus have two spin functions with correct symmetry under the permutation operator
a = c 1 ( +)
b = c 1 ( )
where the electron labels have been suppressed to simplify writing. The value of c 1 is found by normalizing the spin functions. The volume element for two spin variables is d s = d s (1) d s (2); the integrals are factorized; using properties of spin functions in eq 7.6, one finds that c 1 = √. Final results for the two-electron spin functions are collected in Table 7.5.
Table 7.5 Normalized spin functions for two electrons (with electron labels suppressed).
S Spin function MS p
1 1 +
2 - 1/2^ ( +) 0 + - 1 +
0 2 - 1/2^ ( ) 0 - 1
The three functions with S =1 are called the triplet spin functions. They are symmetric with respect to spin exchange. The function with the highest MS =1 value is often referred to as the “high spin” function, and represented by two parallel up arrows in energy level diagrams. The last function in the table with S =0 is called a singlet spin function, and represented by one up arrow parallel to a down arrow. It is antisymmetric with respect to spin exchange.
7.4.2 Two-electron functions of spatial variables
The major problem in solving the electronic Schrödinger equation for the helium atom is the determination of the space part, space (1,2), of the wavefunction. According to case (a) of Table 7.3, a singlet spin function must be multiplied by a spatial function that is symmetric in the exchange of position variables of the two electrons.
(^1) space (1,2) = 1 space (2,1) for S =0 (singlets) eq 7.
where the left superscript indicates the multiplicity of the overall function. On the other hand, a triplet spin function must be combined with an antisymmetric spatial function (case (b) in table 7.3).
(^3) space (1,2) = – 3 space (2,1) for S =1 (triplets) eq 7.
All three triplet spin functions in Table 7.5 are multiplied by the same space function. Since the energy is eventually determined by the space part, one immediately sees that E is independent of the MS quantum number, as stated before.
To proceed further we need to write down explicit expressions for the spatial functions. The usual starting point for this purpose is the “orbital approximation”. If we omit the electron-electron repulsion term from the Hamiltonian in eq 7.1b, we obtain a separable Hamiltonian
̂ ̂( )̂ ( ) eq 7.
where the one-electron Hamiltonian is defined in eq 7.2. The Hamiltonian of eq 7.16 is called the “core” Hamiltonian. It describes the motion of two independent (i.e. noninteracting) electrons in the field of a He nucleus ( Z =2). The one-electron Hamiltonian ̂ is that of a hydrogenlike atom with Z = that we considered in Section 6.3. From eq 6.43, the orbital energies are (in hartree)
eq 7.
The corresponding orbitals are 1s, 2s, 2p 1 , 2p 0 , 2p-1, etc. For the purposes here, it is convenient to employ the complex orbitals because they have well-defined ml values. Both electrons of eq 7. have the same set of energy levels and orbital functions. The only difference in the explicit expressions of the orbitals is the labels of the electrons. For example, 1s(1) and 1s(2) are the 1s orbitals of electron 1 and 2, respectively. Remember that the electron labels denote three spatial variables for each electron.
The eigenvalues of the core Hamiltonian are the sum of orbital energies, and the eigenfunctions are products of orbitals
eq 7.18a
( ) (^) eq 7.18b
Note that the orbital products above are eigenfunctions of ̂ ̂ ̂ with eigenvalues ML = ml (1)+ ml (2). In general, for an orbital product containing N factors, one has
∑ ( ) eq 7.
This is one of the required properties of approximate functions since we know that the exact solutions have the same property, as discussed in Section 7.3.1 above. The spatial parts of the trial functions to be used in the variational integral (eq 7.3a) must also be eigenfunctions of ̂ , and additionally, they must obey eq 7.14 for the singlet states, and eq 7.15 for the triplet states. We will illustrate how this is done by considering the ground state of the He atom, and several of its excited states. Let us first simplify the variational integral a little bit.
7.4.4 Excited electronic states of He atom.
Let us consider the electron configuration 1s2s. It means that we have decided to take the 1s orbital for one of the electrons and the 2s orbital for the other, in eq 7.18b. We may write the product function as either 1s(1)2s(2) or 2s(1)1s(2), since we can not distinguish between the two electrons. Both have ML =0, and L =0; however, they do not obey eq 7.14 or 7.15, required by the Pauli exclusion principle. Hence, we form linear combinations of these two functions to obtain two new functions with desired symmetry properties, as we did for the spin functions above. Suppressing the electron labels, we have
(^1) space = c ( 1s2s + 2s1s) for singlet (S=0) spin state eq 7.
(^3) space = c ( 1s2s - 2s1s) for triplet (S=1) spin state eq 7.
where c=2-1/2^ is a normalization constant. The term symbols of these states are 1 S and 3 S. When the integral in eq 7.20 is evaluated with the functions above, both give higher energies than that of the function in eq 7.21. They form part of the low-lying excited states of He atom.
In general, if u and v are different orbitals, the symmetric linear combination
u(1)v(2) + v(1)u(2) uv + vu for S =0 eq 7.
satisfies eq 7.14 for singlet states while the antisymmetric combination
u(1)v(2) – v(1)u(2) uv – vu for S =1 eq 7.
fulfills requirement 7.15 for triplet states. For simplicity in writing we have omitted the normalization constant c.
Closed and open subshells. The group of orbitals for a given n and a given l is called a “subshell”; there are gl =2 l +1 orbitals in the group. According to the simpler statement of Pauli principle, the maximum occupation number of a single orbital is 2, and hence the maximum occupation number of a subshell is 2 gl. If a subshell in an electron configuration has the maximum occupancy, the subshell
is said to be “closed”; otherwise, it is an “open” subshell. Thus, in 1s^2 configuration, the 1s subshell is closed, and in 1s2s, there are two open subshells. We have seen above (eq 7.21) that the orbitals in the former subshell can give only a singlet space function. Other multiplicities are possible only when there are open subshells in the electron configuration, as in 1s2s which gives both a singlet and a triplet space function (eqs 7.22-23).
There are simple rules for determining the possible values of the total L quantum number from the l values of subshells in an electron configuration. For closed subshells, L =0 (and also S =0). The next rule pertains to 2 electrons in open subshells, as in He. Let l 1 and l 2 be orbital angular momentum quantum numbers of the two subshells (which may be same, i.e. l 1 = l 2 ). Possible values of total L are integers given by
L = Lmax , …, Lmin where Lmax = l 1 + l 2 and Lmin =| l 1 l 2 | eq 7.
It is instructive to consider several examples.
a) The configuration 1s2p gives a single value for L because l 1 =0, and hence L = l 2 =1. Using the orbitals belonging to these subshells, 3 singlet space functions of the form in eq 7.24, and 3 triplet space functions (eq 7.25) can be constructed. The term symbols are 1 P and 3 P.The 1 P term has a spatial degeneracy, (2 L +1), of 3; the 3 P term, on the other hand, has an additional three-fold spin degeneracy, so the total degeneracy of this term is 3x3=9. The total number of states of both spins is 3+9=12.
b) Consider the configuration 2p3p. Here, l 1 =1 and l 2 =1. From eq 7.26 we have: L =2, 1 and 0, designated by D, P, and S, respectively. The spatial degeneracies are 5, 3, and 1, in the same order. The two subshells are different because of their n values. As in this and the previous example; when the two subshells are different, both spin values ( S =0 and S =1) are allowed for each value of L. Thus the terms are: 1 S, 1 P, 1 D for the singlet states, and 3 S, 3 P, 3 D for the triplets. They form an additional set of excited states of He atom with 6 distinct energies. The total degeneracies are 1, 3, and 5 for the singlet energies, and 3, 3x3=9, 3x5=15 for the triplets. The total number of atomic states associated with the 2p3p configuration is thus 36.
c) Next we examine the states from the 2p^2 configuration, which may be written as 2p2p. The latter notation indicates that the two subshells are the same in this case. The possible values of L are as in the previous example: 2, 1, and 0. In the 2p3p case, the orbitals u and v of eqs 7.24-25 are selected from different subshells: u from the set (2p 1 , 2p 0 , 2p-1) and v from (3p 1 , 3p 0 , 3p-1). Hence we can form 3x3=9 different singlet functions (eq 7.24) and 9 triplet functions (eq 7.25), leading to the remarks made in example (b). However here, both of u and v must be selected from the same set: (2p 1 , 2p 0 , 2p-1). Two-electron functions that can be constructed for the singlet and triplet states are shown in Table 7.6. There is only one triplet term: 3 P. The D term is a singlet; there must be 5 functions for L = corresponding to the values of – 2 ML 2. For ML = 1 and 2, there is only one function for each value of ML. They belong to 1 D term. For ML = 0, however, there are two functions. The presence of an extra function with ML = 0 tells us that there must be an additional term with L =0, i.e. 1 S. In conclusion, the allowed terms for the 2p^2 electron configuration are 1 D, 3 P, and 1 S, with three different energies. The total number of states of both spins is 15.^2
(^2) All functions in Table 7.6 except for the two singlets with ML =0 are eigenfunctions of ̂ operator with eigenvalues L ( L +1). By forming linear combinations of the latter two functions, two new functions can be obtained such that one of them has L =2 and the other has L =0.
n electrons are assigned to this subshell, the set of terms arising from the subshell configuration l n
are the same as those from l n max -n. E.g, the term symbols of p^4 are the same as those of p^2 ; a d^9 subshell gives the same term symbol as the one for d^1 , i.e. a single 2 D term (a “doublet”); etc. Note that the term with the lowest energy can always be guessed by Hund’s rule for any permissible value of n for the occupancy of the subshell.
Example 7.1 : What is the term symbol for the ground state of neutral Fe ( Z =26) atom?
Solution : The electron configuration for N =26 electrons in the iron atom is: (closed subshells)3d^6. The 3d subshell here is an open subshell because its n max =10. Hund’s rule is: distribute the 6 electrons into as many different orbitals of the d subshell as possible to achieve a maximum value for MS. There are 5 orbitals in the d subshell. We begin by assigning the first 5 valence electrons with parallel spins (i.e. each with ms =+1/2) to these orbitals, so that the resultant MS is 5(1/2)=5/2, and ML due to them is 2+1+0-1-2=0. The 6th^ electron must be assigned to the d 2 orbital to make the total ML largest; hence ML becomes 2. However, the spin of the added electron must be reversed ( ms =-1/2) because of the Pauli principle, so that MS =5/2-1/2=2.
d 2 d 1 d 0 d-1 d-
MS = 2, ML = 2
There can be only one function of space and spin variables, with highest values for both MS and ML. It follows that S =2 and L =2. Hence, the term symbol with the lowest energy is 5 D (a “quintet”).^3
Exercise 7.4 Find the ground state term symbols for the following atoms: a) oxygen, b) nitrogen, c) bromine, d) argon, e) nickel.
(^3) You will arrive at the same result by taking a d (^4) configuration.
7.7 The total angular momentum J and atomic energy levels
Energies of the “terms” are those of the solutions of the atomic Hamiltonian where the potential energy V takes into account only the Coulomb interactions. When these energies are compared with experimental ones, some qualitatively significant differences are found. As an example of experimental data, Figure 7.2 shows the first four energy levels of the carbon atom. Spectroscopic measurements can be extremely precise as you see from the data in the figure.^4 The first two excited levels are separated from the ground state by only 16.40 and 43.40 cm -1, respectively. On the other hand, separation of the third excited level (denoted by 1 D 2 ), when compared with these values, is very high.
(^1) D
(^3) P
(^1) D 2
16.40 cm -
27.00 cm -
10192.63 cm -
(^3) P 2 (^3) P 1 (^3) P 0
Terms Levels
Ground state
Figure 7.2 The first four experimentally observed energy levels of the carbon atom. The lowest two terms are shown on the left.
What does the theory say about the energy levels of the carbon atom ( Z =6)? Its low-energy terms arise from the configuration 1s^2 2s^2 2p^2. There is only one open shell, namely 2p, and the allowed terms are 3 P, 1 D, and 1 S.^5 We know from Hund’s rule that 3 P has the lowest energy. Numerical calculations show that the next term is 1 D with a separation from 3 P, near 10000 cm -1. On the other hand; experimentally, carbon atom has several closely spaced levels with energies near to that of the (^3) P term. Similar small discrepancies between theoretical energies and experimental data have been
observed for many other atoms.
It has been found that this splitting of a term energy into several levels can be explained by supplementing the atomic Hamiltonian we have been using thus far by a new contribution called the spin-orbit Hamiltonian:
̂ ̂⃗̂ eq 7.
where A is a constant that depends on the particular term (i.e. on L and S ) under consideration. Inclusion of this operator modifies the energy of the term by a correction, ESO , called the spin-orbit interaction energy. In order to find a working expression for ESO , one introduces the total angular momentum operator of the atom. It is a vector sum of spin and orbital angular momenta:
(^4) As an energy unit, 1 cm -1=0.0001240 eV = 0.01196 kJ/mol. (^5) See example (c) in Section 7.4.4.
7.8 Selection rules for transitions between atomic energy levels in absorption
or emission of light.
- S =0; i.e. spin multiplicity does not change in the transition.
- The initial and the final levels must belong to different electron configurations such that there must be a change l = 1 in the subshell of only one electron between the two configurations.
- L =0, 1
- J =0, 1, except that J =0 to J =0 transition is forbidden.
As an example, let us consider absorption of light by ground state, 3 P 0 , carbon atoms. Transitions from this level to J =1 or J =2 levels of the same term, or to the 1 D 2 and 1 S 0 levels of the same electron configuration are forbidden because of rule 2 above. Transitions to triplet states (rule 1) belonging to another electron configuration such as 1s^2 2s2p^3 , (here, l = 1), are allowed within the restrictions of rules 3 and 4.
Exercise 7.5 The electronic configuration 1s^2 2s2p^3 gives rise to 6 terms: 1 D, 3 D, 1 P, 3 P, 3 S, and 5 S.
a) For each of the 6 terms, give the level symbols associated with the term. What is the total number of distinct energy levels? Which of these levels is expected to have the lowest energy? Ans. 10 levels; lowest is 5 S 2.
b) Find the allowed transitions between each of the levels 3 P 0 , 3 P 1 , and 3 P 2 of ground configuration 1s^2 2s^2 2p^2 and the levels of part (a).
