# Vibrational Study and Force Field of the Citric Acid Dimer Based on the SQM Methodology

Authors: Laura Cecilia Bichara; Hernán Enrique Lanús; Evelina Gloria Ferrer; Mónica Beatriz Gramajo; Silvia Antonia Brandán
Journal: Advances in Physical Chemistry
 (2011)

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## Abstract

We have carried out a structural and vibrational theoretical study for the citric acid dimer. The Density Functional Theory (DFT) method with the B3LYP/6-31G∗ and B3LYP/6-311++G∗∗ methods have been used to study its structure and vibrational properties. Then, in order to get a good assignment of the IR and Raman spectra in solid phase of dimer, the best fit possible between the calculated and recorded frequencies was carry out and the force fields were scaled using the Scaled Quantum Mechanic Force Field (SQMFF) methodology. An assignment of the observed spectral features is proposed. A band of medium intensity at 1242 cm-1
together with a group of weak bands, previously not assigned to the monomer, was in this case assigned to the dimer. Furthermore, the analysis of the Natural Bond Orbitals (NBOs) and the topological properties of electronic charge density by employing Bader's Atoms in Molecules theory (AIM) for the dimer were carried out to study the charge transference interactions of the compound.


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## Body

## 1. Introduction

The structural and vibrational studies of citric acid are of chemical, biochemical, and technological interest [1–8]. The crystal and molecular structure of anhydrous citric acid was determined by Nordman et al. [6] and Glusker et al. [7] by means of X-ray crystal analyses. Both authors demonstrated that two molecules of the acid are linked through hydrogen bonds between their carboxylic groups originating the centre-symmetrical cyclic structure of the dimeric species. Many previous studies in relation to the chemical and spectroscopic properties of this acid as monomer have been reported [9–14] while an ab-initio/HF force field study for the vibrations of citric acid and citrate trianion was published by Tarakeshwar and Manogaran  [15]. In the latter study, the calculated geometries and vibrational frequencies only for the monomer citric acid were used to assign the vibrational spectra. A complete vibrational characterization of this acid is very important to obtain information about chemical and physical components in food, biological materials, and pharmaceutical products at the molecular level [14, 16]. So far, there are no publications about experimental or high-level theoretical studies on the geometries and force field for the dimer citric acid. Hence, obtaining reliable parameters by DFT theoretical methods is an appealing alternative to have a precise knowledge of the normal vibration modes. The aim of this paper is to report a DFT theoretical and experimental study of dimer citric acid by infrared spectroscopy in order to effect a complete assignment of every observed bands in the vibrational spectra by taking into account the presence of the dimeric species in the solid phase. For this purpose, the optimized geometries and frequencies for the normal vibration modes, considering the dimer with two units of the free acid, were calculated. For a complete and reliable assignment of the compound, the DFT calculations were combined with the SQMFF methodology [17–19] in order to fit the theoretical frequencies values to the experimental ones. In addition, the force constants were calculated and compared with those reported for the monomer by using the HF/4-21G method [15]. Here, we demonstrated that a DFT molecular force field for the dimer calculated by using the B3LYP/6-31G* combination is well represented. DFT normal mode assignments, in terms of the potential energy distribution, are in agreement with those obtained from the normal coordinate analysis. The total energy calculated with the 6-31G* basis set for citric acid dimer was corrected for basis set superposition error (BSSE) by the standard Boys—Bernardi counterpoise method [20]. Besides, the dimer electronic properties  were evaluated by NBO [21–24] and Atoms in Molecules (AIMs) [25, 26] studies in order to analyze the nature and magnitude of the intermolecular interactions.


## 2. Experimental and Theoretical Calculations

A pure anhydrous Mallinckrodt commercial sample of citric acid (C6H8O7) (CA) was used. The FTIR spectrum of the solid substance in KBr pellets was recorded with a Bruker IFS 66 FTIR-spectrophotometer from 4000 to 400 cm−1. FT-Raman spectrum was also collected on a Bruker IFS 66 FT-IR spectrophotometer provided with a NIR Raman attachment equipped with an Nd:YAG laser at 1064 nm laser.The starting point for the dimer geometry optimization was a structure with bond lengths taken from the X-ray crystal structure [6, 7]. These initial geometries were optimized by using the hybrid B3LYP method [27, 28] with the 6-31G* and 6-311++G** basis sets, as implemented in the GAUSSIAN 03 program [29]. The atom structure and labelling can be seen in Figure 1. The electronic charge density topological analysis was performed by using the Atoms in Molecules (AIMs) methodology [25], by the AIM2000 program package [26] while the NBO calculations were performed by using the NBO 3.1 program [24], as implemented in the GAUSSIAN 03 package [29]. The harmonic wavenumbers and the valence force fields in Cartesian coordinates were calculated at the B3LYP/6-31G* and 6-311++G** levels of approximation. The resulting force fields were transformed into “natural” internal coordinates by using the MOLVIB program [30, 31]. The natural internal coordinates are presented in Table  S1 of the Supplementary Material at doi/0.1155/2011/347072 and have been defined as proposed by Fogarasi et al. [32]. In this case, some coordinates are slightly different from those used in the previous study [15]. The six inter-monomer coordinates for this dimer are similar to those defined by other authors for other dimeric acids [33–36], which can be seen in Supplementary Table  S2. Following the SQMFF procedure [37, 38] the harmonic force fields were evaluated at B3LYP/6-31G* level. In the scaling procedure the diagonal force constants associated with a particular symmetry coordinate are multiplied by a small number of scale factors (fi,fj, etc.). As a result, each off-diagonal constant is automatically scaled by the geometrical mean of factors (fi×fj) 1/2 for the corresponding diagonal constants. Finally, the scale factors are adjusted by an iterative method until they reproduce the experimental wavenumbers as possible. In this case, we use the Rauhut and Pulay scale factors [17, 18]. The potential energy distribution (PEDs) components  higher than or equal to 10% are subsequently calculated with the resulting SQM. The nature of every vibration modes was carried out by means of the GaussView program [39].Figure 1
Theoretical structure of the citric acid dimer. The H-bonding is indicated by dashed lines.


## 3. Results and Discussion

### 3.1. Geometries Optimization

Supplementary Table  S3 shows the total energies and the corresponding dipole moment values for the dimer structures with the B3LYP method by using 6-31G* and 6-311++G** basis sets compared with the ones corresponding to two monomer units. The energy difference between the cyclic dimer and the two citric acid isolated monomers with the B3LYP/6–31G* method is 82.86 kJ mol−1 while by using the 6-311++G** basis set the value at 68.64 kJmol−1 decreases. Reva and Stepanian [40] predict an energy difference of 86.00 kJ mol−1 between the cyclic dimer and two isolated monomers of benzoic acid (ab initio/4-31G) while the calculated energy difference between their monomers is 40.12 kJ mol−1. In this case, the total energy for the citric acid dimer is smaller than the sum of the energies of each monomer (Supplementary Table  S3) by using both basis sets. Such observation might mean that the dimer presence even in the gas phase is preferable to the isolated monomer, as observed in the 4-hydroxybenzoic acid dimer [36]. This result is very important to carry out the assignment of the vibrational spectra of this acid. In general, our theoretical values are in agreement with the experimental values for the citric acid [6, 7]. The dimer stability was investigated by means of NBO [21–23] and AIM calculations [25]. In the NBO analysis, the second-order perturbation energies E(2) (donor → acceptor) involving the most important delocalization for the dimer clearly reveal the energies and occupancies of the main contributions: LP(2)O19 →σ*O22-H21 (115.2 kJ mol−1), LP(2)O27 →σ*O19-H20 (38.4 kJ mol−1) and LP(2)O27 →σ*C3-H6 (2.4 kJ mol−1) to the delocalization energy due to the OH–O bonds between both structures. In Bader’s topological analysis [25] the localization of the Bond Critical Point (BCP) and Ring Critical Point (RCP) in the ρ(r) and the Laplacianvalues ∇2ρ(r) at these points is important to characterize molecular electronic structure. The critical point has the typical properties of the closed-shell interaction when the value of ρ(r) is relatively low, the relationship |λ1|/λ3 is <1, and ∇2ρ(r) is positive indicating that the interaction is dominated by the charge contraction away from the interatomic surface toward each nucleus. The topological properties for the citric acid dimer can be seen in Supplementary Table  S4. The values obtained clearly reveal the formation of hydrogen bonds between two COOH groups of each monomer. In addition, the seven RCPs calculated by means AIM analysis for the dimer corroborate those contributions found in the NBO study.


## 3.1. Geometries Optimization

Supplementary Table  S3 shows the total energies and the corresponding dipole moment values for the dimer structures with the B3LYP method by using 6-31G* and 6-311++G** basis sets compared with the ones corresponding to two monomer units. The energy difference between the cyclic dimer and the two citric acid isolated monomers with the B3LYP/6–31G* method is 82.86 kJ mol−1 while by using the 6-311++G** basis set the value at 68.64 kJmol−1 decreases. Reva and Stepanian [40] predict an energy difference of 86.00 kJ mol−1 between the cyclic dimer and two isolated monomers of benzoic acid (ab initio/4-31G) while the calculated energy difference between their monomers is 40.12 kJ mol−1. In this case, the total energy for the citric acid dimer is smaller than the sum of the energies of each monomer (Supplementary Table  S3) by using both basis sets. Such observation might mean that the dimer presence even in the gas phase is preferable to the isolated monomer, as observed in the 4-hydroxybenzoic acid dimer [36]. This result is very important to carry out the assignment of the vibrational spectra of this acid. In general, our theoretical values are in agreement with the experimental values for the citric acid [6, 7]. The dimer stability was investigated by means of NBO [21–23] and AIM calculations [25]. In the NBO analysis, the second-order perturbation energies E(2) (donor → acceptor) involving the most important delocalization for the dimer clearly reveal the energies and occupancies of the main contributions: LP(2)O19 →σ*O22-H21 (115.2 kJ mol−1), LP(2)O27 →σ*O19-H20 (38.4 kJ mol−1) and LP(2)O27 →σ*C3-H6 (2.4 kJ mol−1) to the delocalization energy due to the OH–O bonds between both structures. In Bader’s topological analysis [25] the localization of the Bond Critical Point (BCP) and Ring Critical Point (RCP) in the ρ(r) and the Laplacianvalues ∇2ρ(r) at these points is important to characterize molecular electronic structure. The critical point has the typical properties of the closed-shell interaction when the value of ρ(r) is relatively low, the relationship |λ1|/λ3 is <1, and ∇2ρ(r) is positive indicating that the interaction is dominated by the charge contraction away from the interatomic surface toward each nucleus. The topological properties for the citric acid dimer can be seen in Supplementary Table  S4. The values obtained clearly reveal the formation of hydrogen bonds between two COOH groups of each monomer. In addition, the seven RCPs calculated by means AIM analysis for the dimer corroborate those contributions found in the NBO study.


## 4. Vibrational Analysis

The dimer’s structure hasC1 symmetry and 120 vibrational normal modes where all vibrations are IR and Raman active. The recorded infrared spectra for the compound in the solid phase can be seen in Figure 2. The assignment of the experimental bands to the 120 expected normal vibration modes  was made on the basis of the Potential Energy Distribution (PED) in terms of symmetry coordinates by using the B3LYP/6-31G* data and taking into account the previous assignment for the monomer [15] and for related molecules [33–36, 40]. Table 1 shows the observed frequencies and the assignment of the vibrational normal modes while the PED based on the 6-31G* basis set is shown in Supplementary Table  S5. The theoretical calculations reproduce the normal wavenumbers by using the 6-31G* basis set, with an initial rmsd value of 11.5 cm−1 while when the SQMFF method is applied by using Pulay’s scaling factors, the final value is 1.5 cm−1. The best results are obtained with a B3LYP/6-31G* calculation because the used transferable scale factors are only defined for this method. The SQM force fields for the compound dimer can be obtained at request. The broad band observed between 3400 and 2200 cm−1 in the experimental IR spectra of the compound in the solid phase can be attributed to the O–H hydrogen bond formed by the spatial arrangement of molecules in the lattice crystal [41, 42]. Note that the theoretical infrared spectra for the monomer structure demonstrate a little agreement with the experimental spectrum (Figure 2), especially in the high wavenumbers cm−1 region. On the other hand, in the theoretical Raman spectrum shown in Figure 3, in relation to the experimental one, a higher intensity in the bands attributed to the OH stretching is observed. The existence of molecules in the solid phase as dimer justifies such a difference because this species restricts the motion of the COOH groups and is strongly affected by the double hydrogen bond, as was also observed in the 4-hydroxybenzoic acid in solid state [36]. Thus, the wavenumbers in the dimer are shifted with respect to the monomer and the splitting of the antisymmetric and symmetric modes is particularly large. In this compound, both monomer and dimer structures are present in the solid phase and, for this reason, a comparison between the average calculated infrared spectra from B3LYP/6-31G* level for the monomer and dimer by using average wavenumbers and intensities with the corresponding experimental ones demonstrates a good correlation, especially in the high wavenumbers region. The discussion of the assignment of the most important groups for the compound (see Table 1 and Supplementary Table  S5) is presented as follows.Table 1
Observed and calculated wavenumbers (cm−1) and assignments for citric acid dimmer.
ExperimentalaMonomerbDimeraIRRamanIRRamanAssignmentSQMcAssignmenta3535 vwν O–H3537ν O33–H393535 vwν O–H3527ν O3–H43498 sν O–H3527ν O19–H213480 vwν O–H3525ν O40–H423446 m3446 vw3482ν O13–H143350 sh3342 vw3470ν O31–H353291 s3276 vw3066O22–H243247 sh3040 sh3035 sh3012νa CH2 op2994 w2996 w3000νa CH23009νa CH2 op2994 w2996 w3006νa CH2 ip2994 w2996 w2975νa CH23003νa CH2 ip2982 m2952νs CH2 ip2982 m2950νs CH22951νs CH2 ip2982 m2934 (21)νs CH22946νs CH2 op2950 w2961 w2945νs CH2 op2933 w2929 m2942O10–H121756 vs1786νs C=O11745 sh1750 w1783νs C=O21735 s1775νa C=O21735 s1771νa C=O11708 vs1708 w1706 s1736 (10)ν C=O1726νa C=O31698 sh1693 vs1695 sh1693 (11)ν C=O1677νs C=O31493 vs1483δs COH1469 w1463νa C–C21430 m1432 m, br1427 s1425 mδCH2,1447ν C29–C30δCH31430 m1432 m, br1430δCH2 ip1424 sh1426δCH2 ip1420 sh1417δCH2 op1413 w1412δCH2 op1389 m1391 s1389 s1389 (9)δ COH,1402δa COHwag CH31389 m1385wag CH2 op1389 m1383δC8–O13–H141365 w1381wag CH2 ip1358 w1350 w1350 wτCH21352ρCH2 ip1340 sh1347 w1325 vwwag CH21334δs COH1308 w1308δs COH1308 w1306wag CH2 ip, δC38–O40–H421308 w1301 vw1303δa COH1292 w1291 vw1303ρCH2 ip, wag CH2 ip1285 vw1280 wδ COH,1289ρCH2 opwag CH31256 vw1266νa C–O1242 m1258νs C–O1214 m1229 w1219ρCH2 ip1214 m1211 m1210 w1205 sτCH21212ρCH2 op1199 sh1201 w1195 wν C–O1174 s1179 w1175 sδ COH1169ν C30–O331167 w1155ν C18–O191140 s1141 s1141 (3)ν C–O1145ν C38–O401140 s1140 w1140ν C1–O31140 s1130 w1127ν C8–O131081 w1083 m1090 w1080 sν C–C1112δa COH1053 m1053 m1053 (6)ν C–C1059ν C8–C5, ν C8–C151051 w1057ν C29–C32, ν C29–C261036 sh1039wag CH2 op1036 sh1036τwCH2 op966 vw926νs C–C2945 w943 vs943 m943 (20)ν C–O911νs C–C1914 sh886τwCH2 op904 w904 m885τwCH2 op904 w904 m900 mν C–C872ν C29–O31881 w886 w871τwCH2 ip867 vw865τ(O22–H24), τ(O10–H12)850 vw857ν C32–C38842 vw842νa C–C1842 vw827γa  (OH–O)#807 vw800δCCC op795 sh810 w792γCOO2781 s784 vs786 (29)ν C–C783δCOO2729 sh723 w735δCOO5700 w685δCOO4686 w687 (6)γCOO674γCOO5666 vw672γCOO3656 sh660 sh655γCOO2 , τ(O3–H4),τwCH2 op640 w642 w645 w645 wτ(OH)637τ(O40–H42)627 vw631τ(O3–H4)627 vw625τ(O19–H21)627 vw621δCOO3599 s597 w598 s598 (5)τ(OH)616τ(O40–H42), δCOO5,δC31–C29–C30, ρCOO5574 w574δCOO6571 w566γCOO4550 w559 m556 (7)τ(OH)544ρCOO2541 w537δCOO1520 sh528 vw519γCOO6514 w499γCOO3 , τwCH2 ip,τ(O19–H21)504 w509 sh503 w496γCOO1492 vw490τ(O33–H39)477 sh476 vw485γCOO6438 vvw453 w412ρCOO4415 w420 wδCOO398τ(O31–H35)397 w392 s383 (11)τ(OH)390δC31–C29–C26382 m387τ(O13–H14)367 w373ρCOO3, ρCOO1367 w358bδ COO368ρCOO5348 w334bδ COO345ρCOO6323 vw320bδ CCO329δC13–C8–C5306 w294δC13–C8–C9306 w271δC26–C29–C32266 w251δC31–C29–C30254 sh255bρCCC236δC5–C8–C15247 vw245bδ CCC222δC9–C8–C5237 vw223bδ CCC221δC31–C29–C32217 w187bδ CCC180δC26–C29–C30212 w174δCCC op140b148νs(O–H–O)#127τwCOO3122δ(OH–O)#118τwCOO5105bδ CCC108νa(O–H–O)#, δCCC ip99τwCOO588bτwCOO92δCCC ip68bτwCOO72τwCC ip57τCC253τwop#44bτwCOO46τwCC op39bτwCOO38τwCOO132τwCOO630τwCOO226τwCOO1, τwCOO2,τwCOO325τCC111γs  (OH–O)#8τwCOO4ν: stretching; δ: scissoring; wag: wagging; γ: out-of plane deformation; ρ: rocking; τ: torsion, τw: twisting; a: antisymmetric; s: symmetric; s: strong; m: medium; w: weak; v: very; sh: shoulder; br: broad.aThis work.bCalculated by HF/4-21G method from [15].cTheoretical values from SQM/B3LYP/6-31G* calculations for citric dimer acid.#Intermonomer coordinates.Figure 2
Comparison between (a) the experimental infrared spectrum in solid state, (b) the calculated infrared spectrum from B3LYP/6-31G* level for the monomer, and (c) the calculated infrared spectrum obtained from B3LYP/6-31G* level using average frequencies and intensities for both monomer and dimer species.Figure 3
Comparison between (a) the calculated Raman spectrum from B3LYP/6-31G* level for the monomer, (b) the calculated Raman spectrum obtained from B3LYP/6-31G* level using average frequencies and intensities for both monomer and dimer species and (c) the experimental Raman spectrum in solid state.


### 4.1. Band Assignments

OH Modes
In the previous assignment for the monomer the theoretical HF/4-21G calculations [15] predict that the behaviour of the central COOH group is different from the terminal groups, as in our case by using the B3LYP/6-31G* level and, as a consequence, the central OH stretching mode belonging to that central group is calculated at higher wavenumbers than the other ones. In the dimer, those modes are assigned according to the SQM calculations; for this reason, the OH stretching modes are associated with the Raman band at 3535 cm−1, the IR bands at 3498, 3446, 3291, and 2933 cm−1, and the shoulder located at 3350 cm−1. The C–OH scissoring mode belonging to the COOH groups involved in the H-bonding formation is calculated at higher wavenumbers than the other ones (1483 and 1402 cm−1) probably because they are coupled with the corresponding C–O stretching modes; for this reason, both modes are assigned to the bands at 1493 and 1389 cm−1. The remaining C–OH scissoring modes are assigned to the shoulder and IR bands at 1340, 1308, and 1081 cm−1, respectively, as seen in Table 1. In the region of lower wavenumbers, the OH torsion modes are expected and these modes are also affected by the presence of H bonding. Previously, those modes in the monomer were assigned, according to HF/4-21G calculations [15], to the bands at 645, 598, 556, and 383 cm−1 while by using the B3LYP/6-31G* level they are calculated between 655 and 387 cm−1; for this reason, the OH torsion modes are associated with the Raman bands at 642, 627, 492, and 415 cm−1.CH2 Modes
These modes in the monomer were previously observed in the 3000–2900 cm−1 region [15]. In this case, the out-of-phase and in-phase antisymmetric stretching modes of these groups can be assigned to the shoulder and IR band at 3035 and 2994 cm−1, as indicated in Table 1, while the out-of-phase and in-phase symmetric modes are assigned to the weak IR band at 2950 cm−1 and to the medium intensity Raman band at 2982 cm−1. In the monomer, the scissoring modes were assigned at 1427 cm−1 while for the dimer those modes were also clearly calculated in this region; thus, they are assigned to the bands at 1430, 1424, 1420, and 1413 cm−1. In the above-mentioned study [15], the wagging modes were associated with the very weak IR band at 1325 cm−1. Here, as predicted by calculation, the bands at 1389, 1365, 1308, 1292, and 1036 cm−1 are assigned to the wagging modes. Previously, in the monomer the twisting and rocking modes were assigned at 1350 and 966 cm−1, respectively [15]. In the dimer, the IR bands located at 1358, 1292, and 1214 cm−1 and the Raman band at 1285 cm−1 are associated with the rocking modes, while the expected twisting modes are associated with the IR bands at 904 and 881 cm−1 and with the shoulder in the Raman spectrum at 1036, 656 and 514 cm−1.COO Modes
In the previous studies for the monomer [15] the band at 1706 cm−1 and the shoulder at 1695 cm−1 were assigned to the C=O stretching modes. Note that the band at 1706 cm−1 (Table 1) was related to the strong Raman band at 1735 cm−1 and assigned to the same mode in spite of the difference between them (19 cm−1). In the dimer, the six expected C=O stretching modes are clearly predicted by SQM/B3LYP/6-31G* calculation at 1786, 1783, 1775, 1771, 1726, and 1677 cm−1. On the other hand, the IR spectrum clearly shows the presence of more than one band in this region; thus, two very strong bands at 1756 and 1708 cm−1 can be seen, each band presenting a shoulder at 1745 and 1698 cm−1, respectively. Therefore, the IR and Raman bands between 1756 and 1698 cm−1 are clearly assigned to those modes. In this way, those shoulders should be also associated with other C=O stretching modes, and so this fact would confirm the presence of the dimer in the solid state. In both HF/4-21G [15] and B3LYP/6-31G* calculations, the C–O stretching modes are coupled with the C–OH scissoring modes. In the monomer [15], those modes were assigned to the IR bands at 1195, 1141, and 1108 cm−1, while in the dimer, the IR bands at 1242, 1174, and 1140 cm−1 and the medium intensity Raman band at 904 cm−1 are assigned to these stretching modes. Previously, the medium intensity band in the IR spectrum at 1242 cm−1 was not observed and therefore not assigned. Taking into consideration their relative position, intensities, and the splitting predicted by calculation, the expected out-of-plane deformation modes (γCOO), in-phase and out-of-phase modes, were predicted as pure modes between 792 and 485 cm−1 and coupled with other modes at 655 and 499 cm−1; for this reason, they were assigned to the bands at 807, 795, 686, 574, and 492 cm−1, as observed in Table 1. In the previous study [15], only one deformation mode (δCOO) was assigned to the band at 420 cm−1. Here, as predicted by calculation, those expected deformation modes for the dimer are assigned to the IR band and shoulder at 781 and 789 cm−1, respectively, and to the Raman bands at 700, 627, 574, and 541 cm−1. The rocking modes (ρCOO) for the monomer were not previously assigned and, in this case, the ρCOO modes were easily assigned to the IR bands at 599, 550, and 438 cm−1 and to the Raman bands at 367 and 348 cm−1. Finally, the twisting (τwCOO) modes for the monomer were only predicted by the HF/4-21G level at 88, 68, 44, and 39 cm−1 and, in our case, for the dimer they are predicted strongly coupled between them and in the low wavenumbers region; this way, they could not be assigned because there are not observed bands in the vibrational spectra in those regions.Skeletal Modes
In the previous assignment [15], the bands in the infrared spectrum at 1090, 1053, 900, 850, and 786 cm−1 were attributed to the C–C stretching modes, while the CCC deformation modes were not assigned because they appear at very low frequencies. Here, the bands in the IR spectrum at 1430, 1051 and 945 cm−1 and in the Raman spectrum at 1469, 1053, 966, 850, and 842 cm−1 are assigned to the C–C stretching modes. The calculations predicted the CCC deformation modes in the lower wavenumbers region; for this reason, they are assigned to the Raman bands between 323 and 212 cm−1 while the two modes predicted at 108 and 92 cm−1 were not assigned.Intermonomer Modes
The intermonomer vibrational modes of the dimer usually appear in the 270–10 cm−1 region [33–36, 40] and they are related to restricted translations or rotations of one molecule against the other (see Supplementary Table  S2). Here, the six intermonomer modes appear strongly coupled with other modes and they are observed in the Raman spectrum with the exception of the γa(OH–O) mode that was predicted by calculations with a higher contribution PED (42%) at 827 cm−1; for this reason, it is assigned to the very weak Raman band at 842 cm−1. Table 1 shows the theoretical assignments for the dimer intermonomer vibrational modes.


## 4.1. Band Assignments

OH Modes
In the previous assignment for the monomer the theoretical HF/4-21G calculations [15] predict that the behaviour of the central COOH group is different from the terminal groups, as in our case by using the B3LYP/6-31G* level and, as a consequence, the central OH stretching mode belonging to that central group is calculated at higher wavenumbers than the other ones. In the dimer, those modes are assigned according to the SQM calculations; for this reason, the OH stretching modes are associated with the Raman band at 3535 cm−1, the IR bands at 3498, 3446, 3291, and 2933 cm−1, and the shoulder located at 3350 cm−1. The C–OH scissoring mode belonging to the COOH groups involved in the H-bonding formation is calculated at higher wavenumbers than the other ones (1483 and 1402 cm−1) probably because they are coupled with the corresponding C–O stretching modes; for this reason, both modes are assigned to the bands at 1493 and 1389 cm−1. The remaining C–OH scissoring modes are assigned to the shoulder and IR bands at 1340, 1308, and 1081 cm−1, respectively, as seen in Table 1. In the region of lower wavenumbers, the OH torsion modes are expected and these modes are also affected by the presence of H bonding. Previously, those modes in the monomer were assigned, according to HF/4-21G calculations [15], to the bands at 645, 598, 556, and 383 cm−1 while by using the B3LYP/6-31G* level they are calculated between 655 and 387 cm−1; for this reason, the OH torsion modes are associated with the Raman bands at 642, 627, 492, and 415 cm−1.CH2 Modes
These modes in the monomer were previously observed in the 3000–2900 cm−1 region [15]. In this case, the out-of-phase and in-phase antisymmetric stretching modes of these groups can be assigned to the shoulder and IR band at 3035 and 2994 cm−1, as indicated in Table 1, while the out-of-phase and in-phase symmetric modes are assigned to the weak IR band at 2950 cm−1 and to the medium intensity Raman band at 2982 cm−1. In the monomer, the scissoring modes were assigned at 1427 cm−1 while for the dimer those modes were also clearly calculated in this region; thus, they are assigned to the bands at 1430, 1424, 1420, and 1413 cm−1. In the above-mentioned study [15], the wagging modes were associated with the very weak IR band at 1325 cm−1. Here, as predicted by calculation, the bands at 1389, 1365, 1308, 1292, and 1036 cm−1 are assigned to the wagging modes. Previously, in the monomer the twisting and rocking modes were assigned at 1350 and 966 cm−1, respectively [15]. In the dimer, the IR bands located at 1358, 1292, and 1214 cm−1 and the Raman band at 1285 cm−1 are associated with the rocking modes, while the expected twisting modes are associated with the IR bands at 904 and 881 cm−1 and with the shoulder in the Raman spectrum at 1036, 656 and 514 cm−1.COO Modes
In the previous studies for the monomer [15] the band at 1706 cm−1 and the shoulder at 1695 cm−1 were assigned to the C=O stretching modes. Note that the band at 1706 cm−1 (Table 1) was related to the strong Raman band at 1735 cm−1 and assigned to the same mode in spite of the difference between them (19 cm−1). In the dimer, the six expected C=O stretching modes are clearly predicted by SQM/B3LYP/6-31G* calculation at 1786, 1783, 1775, 1771, 1726, and 1677 cm−1. On the other hand, the IR spectrum clearly shows the presence of more than one band in this region; thus, two very strong bands at 1756 and 1708 cm−1 can be seen, each band presenting a shoulder at 1745 and 1698 cm−1, respectively. Therefore, the IR and Raman bands between 1756 and 1698 cm−1 are clearly assigned to those modes. In this way, those shoulders should be also associated with other C=O stretching modes, and so this fact would confirm the presence of the dimer in the solid state. In both HF/4-21G [15] and B3LYP/6-31G* calculations, the C–O stretching modes are coupled with the C–OH scissoring modes. In the monomer [15], those modes were assigned to the IR bands at 1195, 1141, and 1108 cm−1, while in the dimer, the IR bands at 1242, 1174, and 1140 cm−1 and the medium intensity Raman band at 904 cm−1 are assigned to these stretching modes. Previously, the medium intensity band in the IR spectrum at 1242 cm−1 was not observed and therefore not assigned. Taking into consideration their relative position, intensities, and the splitting predicted by calculation, the expected out-of-plane deformation modes (γCOO), in-phase and out-of-phase modes, were predicted as pure modes between 792 and 485 cm−1 and coupled with other modes at 655 and 499 cm−1; for this reason, they were assigned to the bands at 807, 795, 686, 574, and 492 cm−1, as observed in Table 1. In the previous study [15], only one deformation mode (δCOO) was assigned to the band at 420 cm−1. Here, as predicted by calculation, those expected deformation modes for the dimer are assigned to the IR band and shoulder at 781 and 789 cm−1, respectively, and to the Raman bands at 700, 627, 574, and 541 cm−1. The rocking modes (ρCOO) for the monomer were not previously assigned and, in this case, the ρCOO modes were easily assigned to the IR bands at 599, 550, and 438 cm−1 and to the Raman bands at 367 and 348 cm−1. Finally, the twisting (τwCOO) modes for the monomer were only predicted by the HF/4-21G level at 88, 68, 44, and 39 cm−1 and, in our case, for the dimer they are predicted strongly coupled between them and in the low wavenumbers region; this way, they could not be assigned because there are not observed bands in the vibrational spectra in those regions.Skeletal Modes
In the previous assignment [15], the bands in the infrared spectrum at 1090, 1053, 900, 850, and 786 cm−1 were attributed to the C–C stretching modes, while the CCC deformation modes were not assigned because they appear at very low frequencies. Here, the bands in the IR spectrum at 1430, 1051 and 945 cm−1 and in the Raman spectrum at 1469, 1053, 966, 850, and 842 cm−1 are assigned to the C–C stretching modes. The calculations predicted the CCC deformation modes in the lower wavenumbers region; for this reason, they are assigned to the Raman bands between 323 and 212 cm−1 while the two modes predicted at 108 and 92 cm−1 were not assigned.Intermonomer Modes
The intermonomer vibrational modes of the dimer usually appear in the 270–10 cm−1 region [33–36, 40] and they are related to restricted translations or rotations of one molecule against the other (see Supplementary Table  S2). Here, the six intermonomer modes appear strongly coupled with other modes and they are observed in the Raman spectrum with the exception of the γa(OH–O) mode that was predicted by calculations with a higher contribution PED (42%) at 827 cm−1; for this reason, it is assigned to the very weak Raman band at 842 cm−1. Table 1 shows the theoretical assignments for the dimer intermonomer vibrational modes.


## 5. Force Field

The force constants for the dimer were estimated by using Pulay et al. [37, 38] scaling procedure as mentioned above. The force constants expressed in terms of simple valence internal coordinates were calculated from the corresponding scaled force fields by using the MOLVIB program [30, 31]. It is interesting to compare the principal force constants, which were collected in Table 2 and calculated at the B3LYP/6-31G* level for some vibrations, with those reported for the monomer of this acid by using the HF/4-21G method [15]. The calculated force constants values for the monomer by using the ab-initio method are slightly different from the values for the dimer by using the B3LYP/6-31G* level, as expected. The HF method overestimates the calculated force constants values due to the fact that it neglects the electron correlation and unharmony. On the other hand, the little variation in the f(C=O) force constant value in the monomer with reference to the dimer is justified because in the latter species two molecules of the acid are linked by hydrogen bonds between their carboxylic groups; therefore, the involved C=O bonds decrease the double bonds character. For this reason, a lower value in the f(νO–H) force constant can be also seen in the dimer in relation to the monomer. Obviously, the force constants related to the CH2 groups in the monomer have lower values than the dimer because in the latter species the number of those groups is higher.Table 2
Scaled force constants for the citric acid dimer.
DescriptionDimeraMonomerbB3LYP/6-31G*HF/4-21Gf (ν O–H)6.616.84f (ν C–C)3.924.34f (ν C=O)12.0613.38f (ν CH2)4.895.35f (ν C–O)6.076.11f (δ CH2)0.750.55f (wag CH2)1.150.65Units in mdyn Å−1 for stretching and mdyn Å rad−2 for angle deformations.aThis work.bFrom [15].


## 6. Conclusions

We conclude the following.(i)
The assignment previously made for the monomer [15] was corrected and completed in accordance with the present theoretical B3LYP/6-31G* results. Also, the assignment of the intermonomer vibrational modes for the dimer is carried out and the assignments of the 120 normal vibration modes corresponding to citric acid dimer are reported.(ii)
Six intense bands in the infrared spectrum at 3498, 3291, 1756, 1708, 1174, and 1140 cm−1 are reported  to characterize the compound dimer.(iii)
Numerous bands of different intensities observed in the vibracional spectra not previously assigned, in this work were assigned to the citric acid dimer.(iv)
The NBO analysis reveals that in the citric acid dimer there are strong intramolecular interactions of charge transfer from O lone pairs to theσ*(O–H) antibonds.(v)
The AIM studies of the charge density for the dimer satisfy the criteria of hydrogen bond interactions.(vi)
The complete B3LYP/6-31G* force field for the citric acid dimer was determined as well as the principal force constants for stretching and deformation modes.



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*Source: 347072-2011-06-20.xml*