Wavefunctions spin, and symmetrization

Many-Electron Spatial Wavefunctions 157 TOOLS OF THE TRADE Photoeiectron Spectroscopy 163 Approximate Solution to the Schrodinger Equation 164 BIOSKETCH Syivia Ceyer 179 Spin Wavefunctions and Symmetrization 179 Vector Model of the Many-Electron Atom 186 Periodicity of the Elements 190 Atomic Structure The Key to Chemistry 191... 

Symmetry dictates that the representations of the direct product of the factors in the integral (3 /T Hso 1 l/s2) transform under the group operations according to the totally symmetric representation, Aj. The spin part of the Hso spin-orbit operator converts triplet spin to singlet spin wavefunctions and singlet functions to triplet wavefunctions. As such, the spin function does not have a bearing on the symmetry properties of Hso- Rather, the control is embedded in the orbital part. The components of the orbital angular momentum, (Lx, Ly, and Lz) of Hso have symmetry properties of rotations about the x, y, and z symmetry axes, Rx, Ry, and Rz. Thus, from Table 2.1, the possible symmetry... 

Because symmetric and antisymmetric properties follow the same rules of multiplication as odd/even or positive/negative multiplication (that is, sym X sym = sym, sym X antisym = antisym, antisym X antisym = sym), all we need to do is identify the symmetry properties of the spatial and spin wavefunctions and combine antisymmetric parts with symmetric parts to get an overall antisymmetric wave-function. The following summarizes the properties, which can easily be verified ... 

Note that the complete wavefunction as written in Eq. (2.47) changes sign if the labels of the electrons (1 and 2) are interchanged. W. PauU pointed out that the wavefunctions of all multielectronic systems have this property. The overall wavefunction invariably is antisymmetric for an interchange of the coordinates (both positional and spin) of any two electrons. This assertion rests on experimental measurements of atomic and molecular absorption spectra absorption bands predicted on the basis of antisymmetric electrOTiic wavefunctirais are seen experimentally, whereas bands predicted on the basis of symmetric electronic wave-functions are not observed. Its most important implication is the Pauli exclusion principle, which says that a given spatial wavefunction can hold no more than two electrons. This follows if an electron can be described completely by specifying its spatial and spin wavefunctions and electrons have only two possible spin wave-functions (a and fi). 

The ground-state wavefunction will be antisymmetric in the spin coordinates of the two electrons and symmetric in their spatial coordinates. It will also have zero orbital angular momentum (an S state) the most general S state can be shown to depend only on the interparticle distances ri, r2, and ri2 [11]. We construct it from a basis of functions of the form... 

The overall symmetry of a given level must be antisymmetric with respect to the permutation P 2 of the two H nuclei to satisfy the Pauli principle. The + and -parity combinations defined by equation (8.202) are antisymmetric and symmetric respectively with respect to P 2 (because the electronic wavefunction has u character, see equation (8.251)). Since the ortho and para nuclear spin states are symmetric and antisymmetric respectively, we see that the + parity states combine with the ortho... 

The symmetric and antisymmetric squares have special prominence in molecular spectroscopy as they give information about some of the simplest open-shell electronic states. A closed-shell configuration has a totally symmetric space function, arising from multiplication of all occupied orbital symmetries, one per electron. The required antisymmetry of the space/spin wavefunction as a whole is satisfied by the exchange-antisymmetric spin function, which returns Fq as the term symbol. In open-shell molecules belonging to a group without... 

If an electronic state corresponds to occupation of a doubly degenerate orbital, say of symmetry r(e), the full square V (e) x T(e) gives the full set of possible symmetries of the space parts of the electronic wavefunction. The square divides into symmetric and antisymmetric components the symmetric component [T(e) ] is associated with singlet states (permutation-ally symmetric space part, permutationally antisymmetric spin part), and the antisymmetric component T(e) with triplet states (antisymmetric space part, symmetric spin part). When the two electrons are in orbitals belonging to different degenerate e pairs, the full set of symmetries in T e) x T e) is accessible as both singlets and triplets. 

By definition, the Hamiltonian of a system of identical particles is invariant under the interchange of all the coordinates of any two particles. The wave function describing the system must be either symmetric or antisymmetric under this interchange. If the particles have integer spin, the wavefunction is symmetric and the particles are called bosons if they have half-integer spins, the wavefunction is antisymmetric and the particles are fermions. Our discussion will be restricted to electrons, which are fermions. 

The even values of J correspond to antisymmetric wavefiinctions and must be combined with symmetric spin wavefunctions. The only allowed rotational states of spin paired, or antiparallel, (t4 ) dihydrogen are thus those with, J = 0, 2, 4... This defines para-hydrogen. There are 27+1 possible spin states for each acceptable J value and it follows from Eq. (6.7) that there is only one spin state for the para-hydrogen ground state, J=0. 

The odd rotational values of J eorrespond to symmetric rotational wavefimctions and, in order to have an antis)omnetric total wavefimction, we must combine these with antisymmetric nuclear spin wavefunctions. The odd rotational states 7= 1, 3, 5... are combined with a symmetric nuclear spin wavefimction (where nuclear spins are parallel, or impaired, tt). This defines ortho-hydrogen and again its degeneracy, in the gas phase, is the 2J+1 possible spin states. 

The Faddeev-Watson equations are suitable to the study of permutational symmetry for identical nuclei. This has been done (Micha, 1974) for the three cases in which (1) C = B (2) C = A" and (3) B = A and C = A", to obtain transition amplitudes for direct, atom-exchange and dissociative processes. Nuclear spin variables were included, and amplitudes were found by successively reexpressing symmetrized amplitudes in terms of unsymmetrized ones, reducing nuclear-spin dependences and uncoupling the equations required for calculations. For example in case (2), the terms in the total wavefunction... 

The origins of symmetry induced nuclear polarization can be summarized as follows as mentioned above molecular dihydrogen is composed of two species, para-H2, which is characterized by the product of a symmetric rotational wave-function and an antisymmetric nuclear spin wave function and ortho-H2, which is characterized by an antisymmetric rotational and one of the symmetric nuclear spin wavefunctions. In thermal equilibrium at room temperature each of the three ortho-states and the single para-state have practically all equal probability. In contrast, at temperatures below liquid nitrogen mainly the energetically lower para-state is populated. Therefore, an enrichment of the para-state and even the separation of the two species can be easily achieved at low temperatures as their interconversion is a rather slow process. Pure para-H2 is stable even in liquid solutions and para-H2 enriched hydrogen can be stored and used subsequently for hydrogenation reactions [54]. 

When we combine the spatial and spin wavefunctions, the overall wavefunction must be antisymmetric with respect to exchange of electrons. It is therefore only admissible to combine a syrrunetric spatial part with an antisymmetric spin part, or an antisymmetric spatial part with a symmetric spin part. The following functional forms are therefore permissible functional forms for the wavefunctions of the ground and first few excited states of the helium atom ... 

Despite the publication of these papers, as was indicated in the introduction, most of the later publications have focused on the calculation of dinuclear complexes employing the broken-symmetry approach proposed by Noodleman et al. In this approach the J value involves the calculation of the energy difference between the high-spin state and a low-spin solution that corresponds to a broken symmetry wavefunction in the case of symmetric homodinuclear complexes. From now on, it will be employed the expressions for the Hamiltonians indicated in Eqs. (1) and (2) Eq. (3) was kept for historical reasons. A general expression, (see Eq. 4) can be proposed for any dinuclear complex using the original broken-symmetry approach proposed by Noodleman [26] ... 

Table 1 illustrates the six wavefunctions that are available in ATOMPLUS along with sample output from solving the Schrodinger equation with these trial wavefunctions for the He atom. Table 1 depicts only the spatial part of the wavefunction. Each spatial wavefunction is symmetric the total wavefunction is the product of this spatial part and the spin part which is antisymmetric. 

To satisfy the Pauli principle the antisymmetric wavefunction must be combined with one of the three symmetric spin states, given by equations (7.26)-(7.28). This particular excited state can therefore exist in three different forms. These have slightly different energies because of the small magnetic interactions which occur between the spin and orbital motions of the electrons, and this causes any spectral lines involving this state to be split into three. For this reason it is known as a triplet state. The symmetric wavefunction, yr, combines with the single antisymmetric spin state, and it is said to form a singlet state. ... 

Notice that the two terms of the spin wavefunction require that electrons 1 and 2 have opposite spin when electron 1 is a, electron 2 is j8, and vice versa. So if nothing else, we ve come up with a fancy way of arriving at a result you already know as the Pauli exclusion principle electrons in the same orbital (the same spatial wave-function) must have different values of m. The spatial wavefunction that puts two electrons in the same orbital if/j is symmetric i/>,(l)i/r,(2). The spin wavefunction in which both electrons have the same m is also symmetric a(l)a(2) or j8(l)/3(2). 

Our unused, symmetric two-electron spin wavefunctions are then aa, t sym> and jSjS. If these are ever to come into play, we need an antisymmetric spatial wavefunction. We ve just found that we can t obtain that for both electrons in the same orbital, but what if we form an excited state of helium, with the electrons in different orbitals Let s take the simplest case, the lowest excited electron configuration ls 2s Now there is a new two-electron spatial wavefunction, and we have the symmetrization problem again. The symmetrized spatial wavefunctions are... 

The spin wavefunction that goes with the symmetric spatial wavefunction lAsym must again be anti> 3 for spatially symmetric ground state. But for lAanti th spin wavefunction may be any of the three symmetric spin functions shown in Eq. 4.36. The number of spin wavefunctions that accompany a particular spatial state is called the spin multiplicity of the state. A state with only one spin wavefunction is called singlet, one with three spin wavefunctions is called triplet, and so on. 

Precise evaluations of the atomic energy for many-electron systems require the sort of explicit analysis we were just looking at, in which the wavefunctions are written out and symmetrized. In a properly symmetrized many-electron wave-function, reversing the labels of any two electrons in the function must change the sign of the function. This leads to complicated wavefunctions for many-electron atoms, particularly because in most cases the spin and spatial parts of the wave-function can no longer be separated. Commonly, matrix algebra is used to determine these wavefunctions in a task we usually leave to computers. 

Write the explicit zero-order wavefunction in terms of all relevant coordinates for the ground state boron atom. Do not normalize or symmetrize the wavefunction, but include the one-electron spin wavefunctions a and jS. 

Write one properly symmetrized wavefunction in terms of and rg for each electron, including the spin wavefunctions a and p, for the lcr 2a- MO configuration in H2. 

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