ESR SPECTRUM OF AMMONIUM FLUOROPHOSPHATE
2487
Electron Spin Resonance Spectrum and Electronic Structure of the PO,"Radical in an X-Ray Irradiated Single Crystal of Ammonium Fluorophosphate
by F. G. Herring, J. H. Hwang, W. C. Lin, and C. A. McDowell Department of Chemistry, The University of British Columbia, Vancouver 8, Canada (Received J a n w r y 6,1966)
The electron spin resonance spectrum of an X-ray irradiated single crystal of ammonium fluorophosphate (NHJ2P03P.H20is most easily interpreted by assuming that the main paramagnetic species formed is the P03*- radical. For this particular case we have derived equations which permit one to calculate the eigenvalues of the appropriate spin Hamiltonian for an arbitrary set of coordinates. From the theoretical expressions developed we calculate by an iterative method that the principal values of the P31hyperfine interaction tensor are 1822.3, 1419.7, and 1408.6 Mc/sec. The probable electronic structure of the POa2- radical is discussed in terms of the semiempirical LCAO extended Huckel molecular orbital method and we have calculated that the eigenfunction for the orbital containing the odd electron is
where
+
+
and qjp3p. are the phosphorus 3s and 3p, orbitals and XoZ8 E (2s' 2s2 2s3), (2p1, 2p2, 2p3,), XOzp, 5 (2p1, 2p2, 2p3,), where the superscripts refer to radical. This eigenfunction leads to theoretical estimates oxygen atoms in the for the principal values of the P31hyperfine interaction tensor, namely, 2088, 1674, and 1674 Mc/sec, which are in reasonably good agreement with the experimental results. XoZp,
4'38
+
+
Introduction The electron spin resonance spectrum of the oriented
P032-radical has been reported in four instances in the literature. 1-4 I n two cases,1*2where detailed calculations were made, the radical was found to occupy special crystallographic positions. It was therefore possible to make esr measurements with the external magnetic field along any principal axes of the A tensor. Because of this, the mathematical treatment was less complicated than in the general case. In the remaining two C L S ~ S where , ~ ~ ~ the radical occupied general positions in the crystal, no detailed calculation of the A tensor was reported. We shall now describe the case in which the P032-radical, formed in an X-ray irradiated single crystal of (NH4)2P03F.H20, occupies a general position in the crystal. With the external magnetic field in a general direction and, in addition, with the hyperfine interaction not much smaller than the Zeeman term,
+
+
the spin Hamiltonian is necessarily more complicated.
A detailed calculation of the A tensor was carried out using a second-order perturbation method for the determination of the eigenvalues. The principal values of the A tensor were found to be smaller than for the cases previously reported but the radical is still believed to be P032-. Experimental Section Single crystals of (NH4)2P03F.H20 were grown from aqueous solutions by slow evaporation. The crystals belonged to the monoclinic system and grew in thick plates. The unique crystallographic axis, b, was found (1) A. Horsfield, J. R. Morton, and D. H.Whiffen, Mol. Phys., 4, 475 (1961). (2) M. W.Hanna and L. J. Altman, J . Chem. Phys., 36, 1788 (1962). (3) J. R.Morton, Mol. Phys., 6,193 (1963). (4) J. R. Morton, J . Phys. Chem. Solids, 24,209 (1963).
Volume 70, Number 8 August 1966
F. HERRING,J. HWANG, W. LIN, AND C. MCDOWELL
2488
'
to lie in the plane containing the flat crystal face. Hence, it coincided with one of the two extinction directions of polarized light in this plane. With the help of esr measurements, the b axis was easily identified. It was therefore natural to choose the remaining extinction direction in this plane as another reference axis which we called a. The c axis was then chosen to be the normal to the plane. The positive sense of b was arbitrarily fixed. The positive senses of a and c were chosen so that a, b, c, in this order, formed a righthanded system. It must be noted that it is immaterial how the positive sense of either a or c is chosen, but, of course, once one is chosen, the other is fixed. The X-ray irradiation and the esr measurement techniques are described el~ewhere.~ Results The esr spectra consisted of two lines for the cases where the crystal was orientated so that HI1 b or H l b and four lines for other orientations. The four lines consisted of two sets of two and were due to two nonequivalent sites and the splitting between two lines of a set varied between 500 and 650 gauss, approximately. The splitting was attributed to the hypefine interaction of the odd electron with the 31Pnucleus in the radical. I n all cases the two sets of lines due to two different sites had slightly different line widths and there was no ambiguity in deciding which two lines belonged to the same set. Figures 1-3 show the angular variation of the splitting between one set of the 31P doublet for orientations corresponding to H l a , H l b , 0 0 the crystal axes.6 The and H l c , where a, b, and c are circles represent the experimental data, and the solid line represents results from calculations based on the theory outlined in the next section of this paper. The Spin Hamiltonian The most general spin Hamiltonian applicable for this case may be written as
+
X = @ H . g * S S - A . 1 - gN@"'I
H l a
Angle of rototion,deg
Figure 1. Angular variation of the hyperfine splitting between one set of doublets for the PS1nucleus in an X-ray irradiated single crystal of (NH&POsF.H20 with Hela. The circles represent the experimental data, and the solid line is the one calculated from the theory outlined in the text.
Hlb
3 3550 l n In 0 ¶
'?
-
U 0)
ii
3500
F
(1)
For 31P, g?;@N/gP = 0.0006; hence, we may neglect the last term in (1) and write instead
+
X = @H*g*S S.A.1
29500
(2)
We shall assume that our g tensor is isotropic; Le., g = gU where U is the unit tensor and we shall further assume that g = 2.000 (Horsfield, et aZ.,l gave the following principal values of g for P03'-: gz = 1.994, gz = 2.0011, gy = 2.0004). Equation 2 then simplifies to
+
X = g@H*S S . A . 1
(3) Let a, b, c, be an arbitrary orthogonal coordinate system taken to be the system described in the previous The Journal of Physical Chemistry
30
60
90
120
150
180
Angle of rototion,deg
Figure 2. Angular variation of the hyperfine splitting between one set of doublets for the PS1nucleus in an X-ray irradiated single crystal of (NH&POsF.H20 with H o l b .
section. Then, for the external field H parallel to a, we have eq 4. ( 5 ) W. C. Lin and C. A. McDowell, Mol. Phys., 7, 223 (1964).
ESRSPECTRUM OF AMMONIUM FLUOROPHOSPHATE
2489
turbation method (first-order corrections were found to vanish) to calculate the eigenvalues. We write Since the hyperfine interaction tensor is nearly isotropic, we may identify a with 2, b with x, and c with y. Then, upon introducing spin shift operators, eq 4 becomes
x = XO+X'
(6)
with Xo = gPHS,
+ AQaSJz + '/4(Abb
+
+ SJ+)
ACC)(#J-
(7)
and X' equal to the remaining terms in eq 5 . The eigenvalue problem for xocan now be solved exactly. The eigenvalues and eigenfunctions are given in Table I where tan 26 =
This last equation can be used cyclicly in the subscripts a, b, c, corresponding to the orientations Hila, Hllb, HIIc, respectively. For H in a general direction, the form of the equation would still be the same if one makes an appropriate rotation of the coordinate axes. A convenient basis set of spin functions to use are of course the strong-field functions Ms,M I ) = 11/2, '/z>, 1-'/2, -'/2), 1-'/2, '/z>, '/2, -'/& With the spin Hamiltonian given in eq 5, this basis set forms a 4 X 4 matrix which unfortunately cannot be factorized. Consequently, we use the second-order per-
I
HLc 36001
I
I
I
+
Abb Acc 29PH
At this point, it is convenient to use an abbreviation. Thus, we write A = A,,, B = Abb, C = Ace,D = Aael E = A,,, F = Aab,and G = {(2gPH)2 ( B C)Z)''r
+ +
P (= sin e cos e) = (B
+ C)(G - PgPH) 2G2 - 4gPHG
The eigenvalues corrected to second order then become
+
P - '/2 (E2 A) = 2Gi 4(gPH A ) (E2 F 2 ) X Eiii,iy = *'/d(G r A ) P '/2 [4(gPi:ifT 2G 4(gPH - A ) f 2G P F2)[4(gPH
*
'/2
-
+
*
*
* 26
]
1
(9)
The resonance condition for the low-field (HI) transition between Ei and Ei, and that for the high-field ( H J transition between Eii and Eiii are given by v)
3 0
W
3 HI1b
0
0
0
2950 0
30
60
90
120
150
180
Angle of rototion , deg
Figure 3. Angular variation of the hyperfine splitting between one set of doublets for the Pal nucleus in an X-ray irradiated single crystal of (NH&POsF.HiO with Hole.
~
where the upper sign goes with i = 1 and the lower sign goes with i = 2.
Calculations Equations 10 and 11 can be combined to form an expression for the component A (= A,,) of the tensor A Volume YO, Number 8 August 1966
F. HERRING, J. HWANG,W. LIN, AND C. MCDOWELL
2490
Table I
A =
f(H2
- HI) + l/4(G2
- GI) +
Table 11: Principal Values and Principal Axes of the A Tensor
+
(B - C)' 4D2 1 16gP [E-kl+
Principal values, yMc/aec----
+ P2)(Gz - 2A) + F2)[4A2 - 4G2A + ( B + C)2
A *z A,, A,,
('/2
(gPH2 - A) - PzG2 4A2 - 8gPHzA - (B C)'
+
('/z
4A2
- Pi)(Gi + 2A)
=
(
*86.5 1790.1 i68.4
(12)
*68.4 1421.3
The relative signs for the elements Aaband Agecannot be determined using the data given above alone. However, it will be shown later that the above choice of combination of signs is the correct one; i.e., goes and - goes with -. The diagonalization of with the above matrix gives the principal values and principal axes of the A tensor shown in Table 11. Our experimental results are compared with earlier data on the P032- radical in Table 111. The values resulting from the present study are smaller than those obtained for the alp hyperfine interaction tensor in P032- for other systems. However, there can be little
+
The Journal of Physical Chemistry
b
C
T0.223 f0.957 f0.186
-0.960 -0.182 -0.214
F0.171 F0.226 f0.959
Table I11 : Principal Values of alp Hyperfine Interaction Tensor for POaz- Radical in Various Crystals
-
+ 4GiA + ( B + C)' (gPHi + A) + Pi4 4A2 + 8gPH1A - ( B + C)' ]
1439.3 f86.5 13.4
(1
-
Similar expressions for the components B and C can be obtained by cyclic permutation of A, B, C and D, E, F. Expressions for components D, E , and F are more complicated to describe and will be omitted. Into these six expressions one can substitute the experimental values of HI and H2 for six orientations together with the assumed g value of 2.000. These are then solved simultaneously by an iteration method using an IBM 7040 computer. The iteration procedure used was found to converge rapidly. The same can be carried out for other choices of six orientations. The averages of all calculations are then taken as the final tensor components which are
A
1822.3 1419.7 1408.6
Direction cosines of the principal axes, wrt
+
Parent crystal
NazHPOB.5H20 MgHP03.6Hz0 ("dzPz06 ("4)zHPO4 (NH&POsF.HzO
Principal values of A , Mc/sec
1967 2210 2420 2070 1822
1514 1730 1960 1600 1420
1513 1730 1940 1600 1409
Ref
1
2 3 4 This work
doubt but that the radical is P032-. Probably there is enough difference in the crystal environment to cause small differences in the electronic orbitals of the radical. The A tensor is also seen to be almost axially symmetric but significantly different from exact axial symmetry. (The over-all experimental accuracy is estimated to be about 1 gauss or 3 Mc/sec). This could again be due to a small difference in electronic structure along two mutually perpendicular directions. To check how well the calculated A tensor predicted the positions of the lines, the final tensor components were substituted back into eq 10 and 11 together with the experimentally determined values of Y, and H2 and HI were calculated for various orientations. The calculated H1 and H2 values were used to draw the curves shown in Figures 1-3. Experimental points are included for comparison. It can be seen that the fit is good. The calculation of the g values using eq 10 and 11 also showed that the g tensor is isotropic to within =+=0.0002.The average g value for all orientations was found to be 2.0006, which is insignificantly different from the assumed value of 2.000. To determine the correct combination of the relative signs for the tensor elements Aab and AB,, a new set of axes was chosen
ESRSPECTRUM OF AMMONIUM FLUOROPHOSPHATE
H II b \
1900 I
Hllc
I60
27 00
Figure 4. Angular variation of the hyperfine splittings for the Pal nucleus in an X-ray irradiated single crystal of (NH4)2POaF.H20with Holu’. The solid theoretical line passes through the experimental data shown m circles only when the correct choice of the relative signs of A,, and A,, is made.
with e’ = c but a‘ and b‘ being 50 and 40” from the a axis, respectively. The crystal was then mounted with a’ vertical in the field and rotated about a‘ through 180”. The splittings measured experimentally were plotted (in open circles) in Figure 4. The solid curves correspond to the calculated splittings for the two sites with the correct choice of the relative signs of A, and Abc. The calculated curve did not pass through the experimental data when the incorrect choice of sign was made. Our estimates of the magnitudes of the principal values of the hyperfine interaction tensor for thealp nucleus lead to a value of 1500.2 Mc/sec for the isotropic part of the A tensor. From the self-consistent field calculations’ the value for the Fermi contact term for the 3s orbital of the 31Pnucleus is 10,100 Mc/sec. The theoretical value for the anisotropic part of A, i.e., (4g@NPN/5h) (v3), for the 3p orbital of phosphorus is 574 Mc/sec. Our experimental data thus lead to a p:s ratio equal to 3.08 for the unpaired-electron orbital. The remaining spin density is distributed over the oxygen atoms in the P032-radical. The Electronic Structure of the P032-Radical That form of molecular orbital theory known as the semiempirical LCAO extended Huckel method has been used with considerable Success for the elucidation of the electronic structure and properties of inorganic molecules.6-8 It is of some interest to see how well this
2491
method can predict the magnitudes of the hyperfine interaction constants of inorganic free radicals. The extended Huckel LCAO approach was first employed by Wolfsberg and Helmholze and was later developed by Hoffman,’ Lohr and Lipscomb,8 and Ballhausen and Gray.9 This requires us to obtain solutions of the matrix equation Hc = EGc, for each of the irreducible representations of the eigenfunction of the symmetry group for the molecule in question. I n this equation, H is the “effective” H-amiltonian matrix, G is the group overlap matrix, and the eigenvectors c, corresponding to the eigenvalues E , can be used as a basis set for the calculation of optical and magnetic properties of the molecule. The off-diagonal matrix elements of H are related to the group overlap matrix, G , by the equation
where the Coulombic integrals, Hi,, are set equal to the negative of the respective valence state ionization potentials (VSIP’s) of an electron in the ith atomic orbital. The VSIP’s of many atoms have been estimated using atomic energy levels.1° The P032- radical has 25 valence electrons and the results of electron spin resonance studies discussed above have shown in accordance with Walsh’s” prediction that this molecular species is probably pyramidal with an OPO angle of about 109”. I n the present calculations we have assumed that the OPO angle has the tetrahedral value and that the P-0 bond length is 1.55 A.12 The atomic orbitals which we have used as a basis set are the phosphorus 3s and 3p orbitals together with the oxygen 2s and 2p orbitals. The symmetry-adapted orbitals for the oxygen ligands can be readily obtained using what are by now standard methods.13 The P032-radical with the structure described above belongs to the point group C3”. The three irreducible representations for the CBVpoint group are A1, A2, and E. The linear combination of oxygen 2p, orbitals belonging to the irreducible representation A2 is non(6) M. Wolfsberg and L. Helmholz, J . Chem. Phgs., 20, 837 (1952). (7) R. Hoffman, ibid., 39, 1397 (1963). (8) L. L. Lot?, Jr., and W. N. Lipscomb, ibid., 38, 1607 (1963). (9) C. J. Ballhausen and H . B. Gray, “Molecular Orbital Theory,” W. Benjamin Inc., New York, N. Y., 1964. (io) J. Hinze and H. H. Jaff6, J . Am. Chem. SOC.,84, 540 (1962). (11) A. D. Walsh, J. Chem. SOC.,2301 (1953). (12) A. F. Wells, “Structural Inorganic Chemistry,” Oxford University Press, Oxford, 1962, p 645. (13) R. McWeeny, “Symmetry-An Introduction to Group Theory,” Pergamon Press Ltd., London, 1963.
Volume 70, Number 8 August 1966
F. HERRING, J. HWANG, W. LIN, AND C. MCDOWELL
2492
bonding, therefore, only two secular equations of the type C c t ( H t j - EGtA = 0
(14)
a
need be solved. The group overlap matrix elements were evaluated using methods previously d e s ~ r i b e dl 4. ~ ~The basic two-center overlap integrals were estimated from Slater atomic orbitals. l5 The required VSIP’s for phosphorus and oxygen were obtained from the tables given by Hinze and Jaffe.lo The valence state for oxygen was taken to be the ~ p , p , ~ p ,state ~ and we employed different VSIP’s for the two types of 2p orbital on the oxygen atom, namely
XOZ~, X02,,
This seemed to be a more satisfactory approximation rather than to assign both of the oxygen 2p orbitals the same value. The iterative technique of Ballhausen and Grays was employed to obtain a self-consistent charge distribution and consequently the VSIP’s of the central phosphorus atom. Application of the aufbau principal to the eigenvalues for the P0s2- radical calculated by the method outlined led to the result that the unpaired electron is in an orbital of AI symmetry. This molecular orbital is a linear combination of the phosphorus 3s and 3p, and is a symmetry-adapted linear combination of oxygen 2s, 2p,, and 2p, orbitals. The explicit form of the odd orbital is
+
where the superscripts number the oxygen atoms.16 This eigenfunction for the odd-electron orbital leads to the estimates for the principal values of the hyperfine interaction tensor for the 31Pnucleus in P032-,which are given in Table IV. Table IV: Theoretical Estimates for t h e Principal Values of t h e Hyperfine Interaction Tensor for *1P in the POP- Radical Spin density on P atom
VSIP(2pJ = -18.53 ev VSIP(2pJ = -15.87 ev
= @pl, + 2p2, + 2p3,) = (2p1, ~ P Z+, 2p3,)
8
P
0.178
0.484
Principal values of hyperfine tenaor, Mc/sec
2088
1674
1674
These theoretical estimates agree well with the experimentally observed values. The calculated p :s ratio is 2.7 which is somewhat less than the expected value 3 but is reasonable in view of the approximations adopted in the calculations. We may thus conclude that this is a satisfactory method for calculating theoretically the magnitude of the hyperfine interaction coupling in inorganic molecular species and that it can be used with confidence as an aid to the interpretation of electron spin resonance spectra.
Acknowledgments. We are much indebted to the National Research Council of Canada and the Defence Research board for generous grants in support of this work. ~~
0.206
-7pnpz
0.635
--
3 z p ,
(15)
where 4p3sand 4p3p, are the phosphorus 3s and 3p, orbitals and xo2,
= (2s’ + 2 ~ 2+ 2~3)
The J O U Tof~Physical Chemistry
(14) C. J. Ballhausen, “Introduction to Ligand Field Theory” McGraw-Hill Book Co., Inc., New York, N. Y.,1962. (15) R. S. Mulliken, C. A. Rieke, D. Orloff, and H. Orloff, J. Chem. Phy8., 17, 1248 (1949). (16) We have used a right-handed coordinate system for the phosphorus atom with the z axis parallel t o the Cg axis and pointing away from the oxygen atoms. A left-handed coordinate system has been used for the ligands, each with the x axis pointing toward the phosphorus atom and the z axis lying in the plane of the ligand x axis and the phosphorus z axis and pointing away from the central atom a
