Wave Functions of Excited States Ally1 Cation, Radical and Anion BY J. W. LINNETT AND 0. SOVERS Inorganic Chemistry Laboratory, Oxford Received 8th January, 1963 Various types of wave functions for the 2, 3 and 4 electrons in the r-orbitals of the allyl cation, radical and anion have been tested and their performance compared. For the cation all nine states whose wave functions can be constructed from the 2pa atomic orbitals on the carbon atoms have been studied, but for the radical only the four 2Az states and for the anion only the four 1Al states have been examined. Full configuration interaction treatments (CI) had been published. The results obtained using a number of approximate functions (molecular orbital, valence-bond and non-pairing) were compared with these (full CI).The non-pairing method gives the best overall agreement with the full CI treatment as judged from calculated energies, overlaps with the CI functions, and charge distribution. The simple non-pairing functions for the different states also required little mixing with one another to achieve orthogonality. " Correlation error " in the calculation of electronic energies is reduced if electrons of opposite spin occupy spatially separated orbitals. Two methods have been used for molecules ; (i) in orbital molecular (MO) theory, " alternant " orbitals,l and (ii) on the lines of valence-bond (VB) theory, spatially separated bond-orbitals ; the " non-pairing " (NP) functions of Hirst and Linnett.2 One object of using different orbitals for electrons of different spin is to remove, with a simple function, a large part of the correlation error for singlet states.Another is to eliminate some of the lack of balance in calculating singlet-triplet separations. Both (i) and (ii) have an advantage over configuration interaction (CI) of giving simply visualizable wave functions. The present object is to see whether the NP method,l which gave good results for the ground states of the n-systems of the allyl cation, radical and anion, may be extended to excited states. Because a complete CI treatment can reasonably be carried out for these 2, 3 and 4-electron n-systems, a necessary requirement for any NP scheme must be simplicity. The main problems are (i) choosing bond orbitals for the excited states and (ii) making the wave functions orthogonal among them- selves; these are not encountered in the MO scheme.If such difficulties can be overcome simply, and if the results resemble those of the complete CI treatment more closely than do those of the MO calculation, it may be preferable to describe excited states in this way. CALCULATIONS The method and notation of Hirst and Linnett 2 will be used for the 2,3 and 4 n-electrons. Full CI results for all states are given by them; our calculations will be compared with this " best " treatment, and with other approximate methods, using the following criteria : (a) energy ; (b) overlap of the trial function with the corresponding " best " one ; (c) charge distribution : bond orders and charge densities ; (d) distribution of the total energy between core-attraction and inter-electron repulsion energies.The last two are much more sensitive tests than (a) and (b). 58J . W . LINNETT AND 0. SOVERS 59 CATJON STATES Let a, b and c be the 2p orbitals on the three atoms, b being the central one. In the ground state, one electron is placed in the bond orbital which is a linear combination of a and b, while the other is in one formed from b and c. A variable parameter k may be introduced in two ways : As in ref. (2) these represent determinants. Combinations of these with equivalent deter- minants must be used to satisfy symmetry and spin requirements. This was done in ref. (2) and the following functions were obtained: aSym = (ka+b, b+kc) or = (a+kb, b+kc). and where and As in ref. (2), the a spin function is associated with the first term and the /3 spin function with the second.(Superscripts (+) have been omitted.) For minimum energy, k = 1.41 for Qym, and 3.97 for mSt. The latter gives a better energy and overlap.2 Six simple n-electron 1Al trial functions can be constructed for the ground and excited states of C3HC if two-centre bonding (B) and antibonding (A) orbitals are used. These are BIB, BB/, A/B, AB/, A/A and AA/. In this notation, A/B represents one anti-bonding orbital using a and 6, and one bonding one using b and c, this determinant being combined with all other equivalent ones to satisfy symmetry and spin requirements : AA/ represents two anti-bonding orbitals on the same pair of centres. Of the above six, only four can be linearly independent, and the most suitable four must be chosen. Then the constants k and k' in bonding and anti-bonding orbitals must be selected.Hirst and Linnett 2 examined BB/ as an approximation for the ground state ; it can be eliminated because it overlaps BIB too greatly. Hence A/B is selected for the first excited state. The two upper ones must be mainly ionic. For this reason A/A is omitted and AB/ and AA/ are retained. The four trial functions B/B, A/B, AB/ and AA/ will be denoted by I q ) in which q = 1,2,3,4. The above set of four simple functions may also be derived from a consideration of the MO configurations. The ground configuration, I MOI), is : I MOI) = (a+kb+c, a+kb+c), = (a++kb, +kb+c)+(+kb+c, a+$kb) (a++% a++kb)+(+kb+c, +kb+c). The first two determinants correspond to BIB, and the last two to BB/, i.e., Similarly, it can be shown for excited states that I MOI) = I BIB) + I BB/).I MOII) = I A/B)+ 1 I MOm> = I A/B) - I D/) 1 MOlv) = I AIA)+I Here I A/B) and I AB/) are combinations of four determinants of the types (a-+kb, +kb+ c) and (a-+kb, a++kb) respectively ; 1 A/A) and I AA/) are combinations of two determinants of the types (a-+kb, +kb- c) and (a-$kb, a-+kb) respectively. The Np function for the ground state is the MO configuration with I BBh dropped; this eliminates the ionic terms on the end atoms. Similarly, if 1 AB/) is dropped from60 WAVE FUNCTIONS OF EXCITED STATES I MOII), these terms are eliminated. The elimination of I A/B) and I A/A) from I MOIJI) and I MOJV) increases the weight of these ionic terms for the two upper states.The constants k and k’ in B- and A-orbitals must now be chosen. For simplicity it is desirable to retain k the same for all B-orbitals in all states. For convenience, the value that minimizes the energy in the ground state will be used. The constant k’ is likewise the same for all states, and is chosen so that k’a- b is orthogonal to a+ kb. This gives k’ = (k+Sab/l+k&), where S,b is the ab overlap. In writing down the state wave functions the symmetric form or the staggered form may be used. Further, two staggered forms (st’, st”) are possible for A/B, AB/ and AA/. In one, the bonding orbital is the one weighted more heavily on the end atom; in the other form, it is the one weighted more heavily on the central atom; the antibonding orbitals are orthogonal to the other bonding orbital in A/B, and to the same bonding orbital in AB/ and AA/.Table 10 in the appendix gives the coefficients Cqj of the basic $j in the functions 4 14) = c CqjI $j>. j = 1 The two lowest states have no contribution from $3 and the two highest states none from $2. Mutually orthogonal wave functions I I), I II), 1 111) and I IV), must now be formed from the 14). The simplest way would be to identify 1 1) with I I), 12) with III} etc., k and k’ being adjusted to achieve orthogonality. However, with only two constants this cannot be achieved between all members of the set and so this approach must be abandoned. Therefore k and k’ were kept the same throughout, as stated above, and orthogonality was obtained as follows.Function I I) was chosen to be NI I l), NI being a normalization constant. For I 11), enough of 1 I) was mixed with I 2) to make the mixture orthogonal to I I). On this basis : A similar procedure is used to make I 111) orthogonal to both I I) and I II), i.e., The function I IV) is formed in the same manner. The method can be summarized as follows. The NP state functions 1 P) (P = I, 11, 111, IV) are given in terms of the basic NP configurations I q) (q = 1, 2, 3,4) by 111) = XI([ 2)-(2 10 11)). I 111) = NIIdI 3)- (3 I 11) I 11)- (3 I 1) I 0). where This is the Schmidt orthogonalization process (these functions are not normalized). The four states may be visualized with reasonable approximation, as single NP config- urations 1 q ) only, if the weights of the admixed functions are small.The main object of this work is to provide accurate descriptions of the excited states within an orbital framework. This is not possible in the CI scheme. To pick the worst example, the CI function for 111 is given in terms of MO configurations by In the simple MO description it was just I MOIII) ; with complete C1, I MOII) and I MOT”) separately outweigh 1 MOIIT) and all simplicity of description is lost. To measure the extent of admixture of lower configurations to the one considered, a quantity, wp, is used. This is given by I 111) = -0.020 I M01)-0°.675 1 MOu)+ 0.471 1 MO111)+0.568 I MOrv). The coefficient of 1 p ) in I P) is unity. The overall admixture w for all four states is taken as the average : IV - . w = * C w p . P = IJ .W. LINNETT AND 0. SOVERS 61 If wtO.1, a good description of the four 1Al states is given by the original NP configur- ations BIB, A/B, AB/, AAI. In contrast, WIII for state I11 in terms of MO configurations, considered above, would be 3.50 (or 1-35 if the levels I1 and I11 are considered to cross over) and the average w for all four states is 1.35 (or 0.84). The hope is that, for many systems as well as allyl, the present procedure will turn out to be preferable to the MO and CI treatments in giving better energies and charge distributions than MOs, while retaining simplicity in having much smaller admixtures of NP configurations (lower w ) than of MO configurations in the CI scheme. Several NP calculations were made for each allyl species. The first involved no para- meters, i.e., k = k’ = 1.When parameters k and k’ are introduced, the configurations p can be chosen in various ways : symmetric (sym) and two staggered (st’, st”). Further, the following configurations were taken as the starting point : the two lower states as st’ and the upper two as st”. This scheme, (st’“t’’), is found to be the best. A further calculation was made, using the above orthogonalization procedure, which may be regarded as the VB analogue, the basic four configurations for the cation being $1, $2, $4, $3, in that order of energy. If the three MOs are a+kb+c, a-c and a-kb+c, k must equal 2/2(1+S,,) for the four configurations to be orthogonal (Sac is the ac overlap integral). A two parameter set of MOs can also be used: a+kb+c, a-c, a-k’bfc, where k’ = 2(1+kSab+S&,k+2Sd) for orthogonality.The energies and charge distributions in this scheme ( k f k’ and k chosen to minimize the energy for the ground state) are practically identical with those of the Huckel scheme. Configuration interaction mixing is also calculated in terms of the two parameter MO configurations for the cation, radical and anion. For judging the functions, the average deviation from 1 of the overlap with the best functions 0 = $XU - Sd, P and the average difference from the best energies, will be used. Table 1 gives the results for w, Q and E for the various treatments of the four 1Al states of the cation. TABLE VALUES OF w, Q AND E OF THE k parameters method - - CI MO 1-56 1.389 VB NP 1 1 NP (st”) 3.974 2.083 NP (st’+ st”) 3.974 2.083 - - NP (sym) 1 -409 1 -222 (in eV) FOR THE FOUR 1Al STATES CATION W I3 e 0.79 (MO) 0 0 0 0-229 2.78 0.124 0.089 0.86 0.247 0.145 1 -46 0.235 0-142 1 -49 0.407 0.252 2-45 0.056 0.009 0.10 MO configurations which have the great advantage of an unambiguous orbital descrip- tion, differ considerably, however, from the best functions (average overlap only 0-771).The VB and simplest NP schemes showed improved overlaps and energies, but use 12 % and 25 % admixture of lower states respectively. The NP (sym) scheme is worse than the VB on all counts, though it gives a better description of the ground state.2 The NP (st”) is the worst of all because, in first excited (A/B) state, it puts too much ionic character on62 WAVE FUNCTIONS OF EXCITED STATES the central atom (ie., gives too much weight to $4).The NP (st’) scheme reverses the energies of levels 111 and IV because it gives$4 too much weight in IV. Because NP (st’) is better for state I1 and NP (st”) is better for state IVY the NP (st’+ st”) gives the best results : an average of only 6 % for admixture of lower states and an average overlap of 0.991. This best sequence of NP (st‘) and NP (st”) states is obtained by the following logical procedure. The best NP(st) wave function for the ground state is constructed. Then the overlap of the two A/B functions NP (st’) and NP (st”) with the GS function are tested ; the results are 0-23 and 0-55 respectively. Because the overlap of the NP(st’) is less it is chosen as the basis for the second (first excited) state.For the third (AB/) state, NP (st’) gives overlaps of 0.83 and 0.19 with the first and second states while NP (st”) gives 0.32 and 0.16. Therefore NP (st”) is chosen as the basis for the third state. By a similar argument NP (st”) is chosen for the highest (MI) state (overlaps of 0-18, 0.09 and 0.07 compared with 0-27, 0-32 and 0-86 for NP (st‘)). The energy for a given state may be divided into W12, the expectation value of the sum of all the inter-electron repulsions (just e2lt-12 for the cation). The core energy W1 will be taken as the difference between the energy of the state (Ep-2Wzp for the cation) and the repulsion energy.2 Table 2 gives the ratio - W12/W1 for all four 1Al states, calculated by the various treatments. TABLE 2.xALCULATED VALUES OF - w12/W1 FOR THE FOUR lAl STATES OF THE CATION (I, 11, 111, IV) CI 0.242 0.267 0.49 1 0.604 MO 0-286 0.435 0-385 0.470 VB 0-249 0.199 0.50 1 0.603 NP 0.276 0.333 0.51 1 0.462 NP (sym) 0.244 0.333 0.537 0.473 NP (st’+ st”) 0.258 0.232 0.49 1 0.61 1 method I I1 I11 IV As expected MO functions make - W12/W1 too large for the two lower states (too little electron correlation) and too small for the two upper states.The orthogonalized VB scheme makes the ratio too small for state 11. Of the three NP methods (st’+st”) is the best and this gives good results. The charge distribution will now be examined. The diagonal elements of the one- particle density matrix are : P where x and s represent space and spin co-ordinates and integration is carried out over the co-ordinates of electron 2.The integral of this over the spin of electron 1 gives the diagonal elements of the spin less density matrixp(x1 I XI). It is p = C,,(ab+ bc)+C,,ac+C,(aa+cc)+C,bb, where c a b , etc., are certain coefficients and a, b and c are functions of the position of electron 1. The quantities cab and C,, are bond orders. Charge densities in the bond regions are then given by S&CUb(Pab) and S,cC,c(pac), and on the atoms by c&,) and CbfPb). The sum, Table 3 lists the charge densitiesp for all four states. The “ deviation” is the sum of the differences from the C1 charge distribution. The table shows that there are large shifts of calculated charges when configurations are allowed to interact ; differences are especially large in states I1 and IV. The NP (st’+st”) functions give much the best charge distribution of the approximate treatments though NP(sym) is much the best for the ground state.The NP (st’+ st”) functions for the four states are represented diagrammatically in fig. 1 which gives some data about each. 2pab + p a + 2pu+pb, equals 2.J . W. LINNETT AND 0. SOVERS 63 TABLE 3.-cHARGE DENSITIES IN UNITS OF THE ELECTRONIC CHARGE FOR FOUR lAl STATES OF THE CATION method STATE I CI MO VB NP NP (st’+ st”) CI MO VB NP NP (syni) NP (st’+ st“) NP (sym) STATE I1 STATE III CI MO VB NP NP (SYm) NP (st’ + st”) STATE IV CI MO VB NP NP (st‘+ st”) NP (sym) P CI energy NP energy overlap wp % Pob 0.24 0.27 0.12 0.24 0.24 0.24 -0.10 0 -0.13 -0.12 - 0.09 -0.12 - 0.26 -0.15 - 0.28 - 0.02 - 0.03 - 0.28 - 0.33 - 0.56 -0.16 - 0.55 - 0.56 - 0.29 I - 30.396 - 30.200 0.988 0 Pac 0.03 0-03 0.03 0.0 1 0.02 0.02 - 0.01 - 0.08 0.00 0.0 1 0.00 0.00 0.0 0.04 0.0 1 - 0.04 - 0.03 0.0 1 0.03 0.06 0.01 0.07 0.06 0.02 I1 CH~--OCH~CH~ CH~-CH~CH~ C H ~ ~ ~ C H - C H ~ CH;-CH-CH~ - 0 - 0 - -22.785 - 22.839 0984 5.3 Pa 0.4 1 0.3 3 0.43 0.29 0.43 0-32 0.82 1 *04 1.06 0-61 0.57 0.95 0.45 0.55 0.14 0.95 0.95 0.42 1 -02 0.78 1.08 0-86 0.75 1.01 P b 0.67 0.79 0.89 0.93 0.65 0.87 0-56 0 0.13 1 no2 1.05 0.33 1.62 1.15 2-28 0.18 0.18 1.71 0.60 1.51 0.16 1 *32 1.58 0.55 111 - 16.247 - 16.313 0996 12.7 deviation - 0.34 0.50 0.52 0.07 0.39 - 1-27 0-98 0.94 1-02 0.54 - 0.93 1.33 2.96 2.93 0.20 - 1.88 0.92 1-52 2.0 1 0.16 IV - 10.460 -10.541 0.997 4.6 FIG.1.-Diagrammatic representation of the NP functions used for the four 1A1 states of the cation.Also listed are the CI and NP calculated energies, the CI/NP overlap and wp for each NP function. (Electrons of one spin are represented by o and of the other by x. If the orbital occupied is an antibonding one a bar is placed over the o or x ; otherwise the orbital occupied is a bonding one. The o or x is placed in the bond nearer to the carbon atom whose 2p7r atomic orbital makes the bigger contribution to the two centre orbital.)64 WAVE FUNCTIONS OF EXCITED STATES STATES There are two 1B2 states, and only functions based on electrons occupying orbitals in the same bond (BB/, AB/, AA/) can have this symmetry. The lower state is BB/ and the function is The value of k (5.92) which minimizes the energy gives the exact CI function.The upper state is AA/ and the function is (k’a- b, k’a- b)- (k‘c- b, k’c- b). If k‘ is chosen so that k’a- b is orthogonal to a+ kb, its value is 2-43. It is then found that to make the upper state orthogonal with the lower state 1.3 % of the latter must be mixed with the above upper state function. It becomes then the CI function exactly. (a+ kb, a+ kb)- (c+ kb, c+ kb). 3& STATES There are two 3B2 states, and only functions based on electrons occupying orbitals in different bonds can have this symmetry ; BIB and A/A will be used. The functions are : BIB : (a+ kb, kb+ c)- (c+ kb, kb+a), A/A : (k’a- b, b- k’c)-(b- k’c, k’a- b). The value of k to minimize the energy of the lower state is 0.807, and that of k’ to make a+ kb orthogonal to b- k’c is 0.882.To make the above upper state function orthogonal to that for the lower one, 12 % of the latter must be mixed with it. 3A1 STATES This state is B/A and any function of this type reproduces the CI function exactly. RADICAL With three electrons the following difficulties arise. There are more orbitals than electrons so that the way they are occupied is uncertain and also, because the electrons are all in different orbitals more than one spin combination is allowed. Nevertheless, reason- able choices of bond orbitals and spin assignments give good approximation to 2A2 wave functions of the ally1 radical. 2A2 STATES The best NP function for the GS of the radical is a resonance hybrid of d H 2 CH-CcH2 and its mirror image.2 The function is made up of determinants of the type (a, a f k b b+kc).The radical GS is related to the cation GS by putting the third electon in the, 2pn atomic orbital on the end atom. For energy minimization k = 3.58 which is en- couragingly close to the value for the cation (3-97). This suggests that the three excited states might be formed from those for the cation by adding an electron to the end atom. Thus the second state would be described by a combination of (a, a+ kb, k’b- c) with other equivalent determinants. The third state would be made up of determinants such as (a, b f k c , k’b-c) and the top state from ones such as (a, k’b-c, k’b-c). Table 11 in the appendix gives of the basic @js (see ref. (2)) in the NP configurations 1 1> to 14). This scheme will be called NP’. This procedure is unsatisfactory because in I 3) and I 4), @4 ((a, a, b)+ (b, c, c)) should have considerable weight.The alternative for I 3) and 1 4) is to add the third electron to A/B and A/A giving I 3) as a combination of determinants like (a, k’c-b, a+kb) and I 4) as a combination of ones like (a, k’a-b, k’b-c). The coefficients in these two functions are also given in table 11. This scheme will be called NP”. Of the various spin arrangements, the one taken is the projection of that which alter- nates apa along the molecule 2 (i.e. (afh-+aap-+pcca)). The use of other combinations would complicate the method. As with the cation, k = 3.58 for all states and k’ is also constant and is chosen in the same way. Table 4 gives the results for w, Q and E for the various methods.J .W. LINNETT AND 0. SOVERS 65 TABLE VALUES OF w, cr AND E (in eV) FOR THE FOUR 2A2 STATES OF THB RADICAL CI - L 0*396(MO) - I parameters k k' method W d e MO 1 -52 1.406 0 0.143 0-96 VB - 0 0.126 0.136 1 -42 NP' 3.584 1 ~990 0.404 0.059 0.63 NP' 3.584 1.990 0.004 0.034 0.3 3 Configuration interaction does not involve as much mixing of states as for the cation but it is still 40 %. Likewise the average overlap of the simple MO functions with the CI functions is only 0.857. This amount of mixing is still too high for the CI description to be regarded as a satisfactorily visualizable one. The orthogonalized VB method gives poor results for the radical. Also the value of w for the NP' functions is much too large. On the other hand, for the NP" functions this is very small indeed and the energies and overlaps are most satisfactory.Because the overlap is 0.999 for the GS and the energy is only in error by 0.01 eV, the values of Q and E for the other three states are 0.045 and 0.43 eV respectively. The values of - W I ~ / Wl for the radical are given in table 5. (Here FV1 = Ep-3 Wzp- W12.) All treatments give similar values for this ratio except that the MO method makes it too high for the GS and too low for the highest. TABLE 5.-cALCULATED VALUES OF - w12/w1 FOR THE FOUR 2A2 STATES OF THE RADICAL (I, II, III, IV) method r II III IV CI 0.470 0.670 0.656 0-786 MO 0.514 0.66 1 0-665 0.734 VB 0.462 0.690 0.656 0.747 NP' 0.469 0.677 0.652 0.772 NP" 0.469 0.677 0.649 0.780 Charge and spin densities were calculated for the radical.The following expression was obtained : p = (C$)u2 + Ci{)p2){a b + bc) + (Cz)ci2 + C$b2)ac + (C$)ci2 + CiB)p2)(aa + cc) 3- where the constants C are functions of &b, Sac and of the coefficients of the basic wave functions. The charge densities are defined as : (CP)a2 + C p p ) b b , in the pob = Sab( cz) + c:;)) ; pa = c f ) + c:'); pac = Sue( cg + @); p b = cp + cp. Spin densities p' were calculated assigning negative signs to the p components, e.g. p; = Cf)- Cf). Defined in this way, 2p&-tp&-2pa+pb = 3, and 2pAb+pA,+2pi+p; = 1. Table 6 gives the results (those for NP' are omitted as they are worse in all cases than those for NP"). For the radical, the results obtained with the different treatments are much more similar to one another. The NP" results are, however, the best except for state IV.The NP" functions for the four 2A2 states are represented diagrammatically in fig. 2 which gives some data about each. ANION lA1 STATES If we consider the following six morbitals along the allyl system: a, kafb, a+kb, kb+c, b+kc, c, the NP function for the GS of the allyl anion can be constructed from C66 WAVE FUNCTIONS OF EXCITED STATES TABLE 6.rHARGE ( p ) AND SPIN (p') DENSITIES IN ELECTRONIC UNITS FOR THE 2A2 STATES OF THE RADICAL method STATE I CI MO VB NP'' STATE II CI MO VB NP" STATE III CI MO VB NP" STATE IV CI MO VB NP" 0.19 -0.01 0.90 0.83 - -0.01 -0.02 0.64 -0.24 - 0.27 -0.01 0.86 0.77 0.30 0 -0.04 0.52 0 0.52 0.06 -0.01 0.97 0.94 0.51 -0.02 -0.01 0.68 -0.31 0.18 0.19 -0.01 0.91 0.80 0.05 -0.01 -0.02 0.64 -0.24 0.00 -0.15 -0.02 0.84 1.64 - -0.20 -0.02 0.55 0.32 - -0.15 0.00 1.07 1.15 0.97 0 -0.04 0.52 0 0.80 -0.21 -0.03 0.62 2.21 1.14 -0.08 -0.03 0.56 0.08 0.51 -0.20 -0.02 0.74 1.94 0.60 -019 -0.01 0.56 0.27 0.10 -0.14 0.02 1.31 0.66 - 0.13 0.00 0.25 0.25 - -0.15 0.00 1.07 1.15 1.01 -0.15 0.04 0-20 0.77 1-22 -0.22 0.04 1.59 0.22 1.18 0.07 -0.04 0.48 -0.07 0.94 -0.21 0.03 1.46 0.46 0.65 0.14 -0.02 0.40 -0.05 0.64 -0.49 0.02 1.25 1.47 - -0.02 -0.04 0.33 0.43 - -0.56 0.02 1.29 1.53 0.28 0 -0.04 0.52 0 0.85 -0.23 0.01 1.11 1.23 1.05 -0.07 -0.00 0.04 1.07 1.36 -0.39 0.01 1.19 1-40 0.40 -0.04 -0.02 0.17 0.75 0.73 those of the cation by filling the " holes " and omitting the electrons from the latter.For the symmetric function this gives (a, a+kb, kb+ c, b) and for energy minimization k = 1.60 (cf.1-41 for the cation). For the staggered function this gives (a, a+kb, b+kc, c) and k = 4.33 (cf. 3-97 for the cation). For the anion the NP symmetric functions for states P I I1 HI IV 0 - - CH2XCHLCHz CI energy - 28.914 - 16.422 - 15.972 - 8.943 NP energy -28.904 -16.144 - 15.608 -9.595 wp % 0 0 2 0-5 0 9 overlap 0.999 0965 0946 0.953 FIG. 2.-Diagrammatic representation of the NP functions used for the four 1A2 states of the radical. Also listed are the CI and NP calculated energies, the CI/NP overlap and wp for each NP function. For symbolism see fig. 1. IT, I11 and IV are (a, a+ kb, b- k'c, c), (a, a, kb+ c, b- k'c) and (a, a, b- k'c, b- k'c) respec- tively, equivalent functions being included in all cases to achieve symmetry.The spins associated with the orbitals are in the order a, p, a, p, and the combination that is the pro- jection of this is used. For the Np staggered functions the following were used for 11,J . W. LINNETT AND 0. SOVERS 67 111, and IV: (a, n+kb, k’c-c, c), (a, a, b+kc, k‘b+c) and (a, a, k‘b-c, k‘b-c). The coefficients of the $j of Hirst and Linnett 2 are given in table 12 in the appendix. The NP staggered scheme is derived from NP (st’) for the cation. Those derived from NP (st”) and NP (st’+ st”) give poor results. Table 7 gives the results for the various functions applied TABLE ST VALUES OF w, CT AND E (ineV) FOR THE FOUR 1.41 STATES OF THE RADICAL W parameters k k‘ method U a CI - - 0.763 (MO) 0 0 MO 1 -48 1 -424 0 0.222 2.74 VB - - 0.123 0.105 0.97 NF ( S Y d 1.599 1.313 0-254 0.124 1 -25 NP (st) 4.325 2.158 0.055 0.020 0.23 to the anion.The general performance of the NP (st’) is better than that of the NP (sym) function, though the latter is better for the GS. For the NP (st’) functions the mean energy deviation calculated is about a twelfth of that obtained using MO functions. If the MO functions are mixed in the CI functions w is 0.763, while for the NP (st‘) functions it is less than a twelfth of this. The ratio - W I ~ / W ~ for all four 1.41 states, calculated using the various methods, is listed in table 8. With the cation and the radical, W12 was the increase in energy due to TABLE 8.-CALCULATED VALUES OF - w121Wi FOR THE FOUR ‘A1 STATES OF THE ANION (I, II, m, W ) method I1 I11 IV CI 0.794 0.899 1.01 1 -09 MO 0-8 10 0.960 0.97 1.06 VB 0.8 14 0,893 1.00 1 *07 Np bym) 0.799 0-907 1.03 1.06 NP (st) 0.797 0.895 1.01 1.09 inter-electron repulsion on bringing the three separated parts together to form the molecules.In the anion, because of the presence of four electrons, W12 is greater than this increase by an amount equal to the inter-electron repulsion energy in the fragment containing two electrons. As before, the MO scheme gives too high a repulsion in the lower states and too low a repulsion in the two upper ones. The VB scheme is better for the excited states, but it gives a poor value for the ratio for the GS. The NP(st’) method gives ratios which are closest to those obtained using CI. The charge densities, defined as before, are given in table 9. The NP (st’) scheme gives the best overall results but the NP(sym) method is better for states I and 11.Probably the most successful NP procedure would be to use symmetric functions for the lower two states and staggered ones for the upper two. However, this would not provide a uniform set as the values of k and k’ would have to be different in I and I1 from those in I11 and IV. The NP (st’) functions are shown graphically in fig. 3, together with some relevant data. DISCUSSION The results of the approximate methods have been compared throughout with those obtained using full CI. The reason for doing this is that the latter gives the best result obtainable when similar restrictions are placed on the wave functions (constructed from 2pn: atomic orbitals) and the Hamiltonian. Any agreement with experiment obtained with an approximate method which is not also obtained using fuli CI would necessarily be fortuitous.68 WAVE FUNCTIONS OF EXCITED STATES TABLE CHARGE DENS^ IN UNITS OF THE ELECXRONXC CHARGE FOR THE lAl STATES OF THE ANION method STATE I CI MO VB NP ( S Y d NP (st') CI NO VB NP (Sym) NP (st') STATE II STATE m CI MO VB NP (st') NP (SYm) STATE IV CI MO VB Np ( S Y d NP (st') P CI energy NP energy overlap wp % Pub 0.22 0.27 0-12 0.19 0-23 - 0.23 - 0.30 - 0.28 -0.12 - 0.26 - 0.30 -0.15 - 0.28 - 0.22 - 0.33 - 0.44 - 0.56 - 0.30 - 0.56 - 0.39 I - 14.506 - 14054 0975 0 Pao Pa - 0.05 1.33 - 0.05 1.39 - 0.05 1-47 - 0.04 1.33 - 0.05 1.46 0.02 1.36 0.09 1.1 1 0.01 1.14 0.0 1 1-57 0.01 1 *22 0.02 1-79 - 0.04 1 -59 0.01 2.14 0.03 1.17 0.01 1 -90 -0.01 1 *42 - 0.02 1.79 0.00 1.15 - 0.02 1.83 0,oo 1.30 11 EHz-kH-CHz OX EH~X~Z'OZCH~ - x X E H z - C H ~ C H ~ - EH2-CH%H2 X X - - 6.48 1 - 6.760 0-973 4 5 P b 0-95 0.75 0.89 1.00 0.66 1.73 2.30 2-28 1.20 2.07 1.00 1.15 0.28 2.06 0.84 2.06 1.56 2.30 1.49 2-17 I11 dev - 0.42 0-54 0.12 0.57 - 1.28 1.10 1.18 0.69 - 0.9 1 1.47 2.47 0.45 - 1 -49 1.07 1.64 0.46 IV + 0.475 + 5.748 +0.477 f5.573 0.986 0988 12.9 4.6 FIG.3.-Diagrammatic representation of the NP functions used for the four 1A1 states of the anion. Also listed are the CI and NP calculated energies, the CI/NP overIap and wp for each NP function. For symbolism see fig. 1. The MO method has the great advantages that the basic set of orbitals is con- structed easily, that it takes account of symmetry straightforwardly and that the wave function for any state is orthogonal to those of all other states (w = 0).For many applications these advantages may outweigh all others. However, when configuration interaction is applied, the simplicity of representation and of visual-J . W. LINNETT AND 0. SOVERS 69 ization is largely lost. These calculations have shown that, with an NP scheme based entirely on uniform sets of two-centre bonding and anti-bonding orbitals, it is possible to obtain energies for the ground and excited states which are very much closer to those obtained with full CI (see fig. 4). The mean improvement for the 1Al states of the cation is 28-fold, for the 2A2 states of the radical is 3-fold and for the 1A1 states of the anion is 12-fold.This is achieved using consistent schemes and orthogonalized wave functions which involve only 5+ %, 3 % and 54 %, ,-* - .. - CATtON RADICAL MO C1 NP MO CI NP -.- .-.. --.-.- -__...-- ---- --- -.. -. . . - - MO CI NP ANION FIG. 4.--Comparison of energy levels calculated using (i) a simple MO treatment, (ii) a full CI treatment, and (iii) the best NP treatment, for the 1A1 states of the ions and the 2Az states of the radical. respectively, of admixture of other states to achieve orthogonality. An orthogonal- ization procedure could have been adopted which modified all states equally (rather than 11, 111 and IV successively) but this would have been much more lengthy and difficult though it would have reduced the above percentages. The calculations here do not, of course, give the highest excited states that exist because the treatments are restricted to combinations of 2p71 orbitals. However, the inclusion of 3p or 3d orbitals might be expected to lead to combinations with NP functions similar to those with CI functions because they resemble them closely. The VB method is found to give quite good results for the cation and anion, both of which involve an even number of electrons. We wish to thank the National Science Foundation for a Fellowship to 0. S., the Royal Society and Imperial Chemical Industries for calculating machines and D. M. Hirst and others for their help. 1 Lowdin, Symp. MuZecuZar Physics (Maruzen, Tokyo, Japan, 1953), p. 13 ; Physic. Reu., 1955, 97, 1509. Pauncz, de Heer and Lowdin, J. Chem. Physics, 1962, 36, 2247, 2257. de Heer, J. Chem. Physics, 1962, 37, 2080; P a w , J. Chem. Physics, 1962, 37, 2739. 2 Hirst and Linnett, J. G e m . Suc., 1962, 1035, 3844.70 WAVE FUNCTIONS OF EXCITED STATES APPENDIX TABLE lO.-COEFFICIENTS OF $j IN EXPRESSIONS FOR THE THREE NP FUNCTIONS FOR THE CATION: Syln, St' AND St" 4 SYm 1 2 3 4 st' 1 2 3 4 St" 1 2 3 4 k 1 - kk' 1 - kk' - k' l+k2 k-k' kk'- 1 - k' l+k2 k- k' kk'- 1 - k' 1 2k 0 0 2k - 2kk' 0 0 2k 2 0 0 c93 0 0 2k 1 0 0 - 2k 1 0 0 2k' k'2 c94 2k2 - 4k' - 4k' 2k'2 4k 4 4k' 2k'2 4k 4kk' - 4k 2 TABLE COEFFICIENTS OF t,hj IN EXPRESSIONS FOR THE TWO Np FUN~IONS FOR THE 4 c91 =92 c93 cq4 1 2k2 3k 3k 3 2 2k 3 - 3kk' - 3k' 3 2kk'- 1 - 3k 3k' 0 4 - 3k' 3 3k'2 0 RADICAL: NP' AND NP" NP' NP" 1 2 3 4 2k2 3k 3k 3 2k 3 - 3kk' - 3k' - 2k 3kk' -3 3k' 2 - 3k' - 3k' 3k'2 TABLE 12.-cOEFFICIENTS OF +j IN EXPRESSIONS FOR THE TWO NP FUNCTIONS FOR THE ANION: SYm AND St' 4 c4.1 %2 cq3 cq4 1 k - k2 2 0 2 kk'- 1 2k 4k' 0 3 kk'- 1 0 4k' - 2k 4 - k' 0 2k'2 1 1 l f k 2 - 2k 4k 0 2 k- k' 2kk' 4 0 3 kk'- 1 0 - 4k 2k' 4 - k' 0 2 k'2 SYm s t'