Exchange Crosspeaks In Noesy

Figure 10.18

Figure 10.18

10.3.4 ROESY: Rotating-Frame Overhauser Effect Spectroscopy

For "medium-sized" molecules where mtc = 1, zero-quantum and double-quantum relaxation rates are nearly the same and the NOE cross-relaxation rate approaches zero. This size depends on the field strength of the NMR instrument and the viscosity of the solvent: it is around 2000 Da for spherical polypeptides in water at 500 MHz. This can happen for peptides, oligosaccharides, and large natural products. Even if the NOE is not zero, it can be too small to conveniently measure with a NOESY experiment. In these cases we have an alternative experiment, originally called CAMELSPIN, which gives negative crosspeaks regardless of molecular size. Instead of the NOESY mixing sequence, which consists of putting the magnetization on the z axis and waiting a period of time for the z magnetization perturbations to propagate to nearby nuclei, ROESY puts the magnetization on a specific axis in the x'-y' plane and "locks" it there for a period of time (the mixing time) using a continous-wave spin lock. The pulse sequence (Fig. 10.18) is almost identical to the 2D TOCSY: We start with the standard homonuclear 90°x-t 1 sequence and then select the in-phase coherence on the x' axis by executing a long, low-power radio frequency pulse.

ROESY spin lock(xr)

We saw in Chapter 8 that the spin lock causes magnetization to transfer in a through-space manner very similar to the NOE, except that now it is x' magnetization on Ha transferring to x' magnetization on Hb (if the spin lock is applied on the x' axis): IX ^ — IX. The main difference is that the effective field felt by the spins is reduced from the static field (Bo) to the radio frequency field strength (Beff), which is typically five orders of magnitude (10—5 times) lower. It is as if we could use our normal magnetic field strength (e.g., 500 MHz) for the preparation, evolution, and detection periods, but switch the field to a very low field strength (e.g., 3300 Hz) for the mixing period. In this low field environment, the SQ frequency is 3300 Hz and the DQ frequency is 6600 Hz. The dominant pathway for relaxation is now Ha(^)Hb(^) ^ Ha(a)Hb(a) (double quantum relaxation), regardless of the molecular size. So we see negative NOE crosspeaks relative to the positive diagonal for large, medium, and small molecules.

The ROESY used to be a bit difficult to set up because the low power spin-lock RF had to come from a different source than the hard pulse RF. Now rapid solid-state power switching is so routine that all1H RF comes from the same source, with no variation of phase or frequency. ROESY tends to replace the NOESY experiment for NOE measurements, especially for small molecules where T2 is relatively long. Because the NOE builds up about twice as fast in the x'-y' plane as it does on the z axis, ROESY mixing times are set to about half of what would be the NOESY mixing time. There is one additional parameter to set up: the power level ("B1 field strength") of the spin lock pulse. This is typically

Figure 10.19

calibrated to give a 90o pulse of 75 |xs (yB1/2n = 3333 Hz) to "cover" a spectral width of about twice that amount (6666 Hz). At this power level, the spin-lock axis is tilted out of the x'-y' plane by an angle of 45o for peaks at the edges of the spectral window.

10.3.5 Examples of NOESY and ROESY

The upfield region of the 600-MHz NOESY spectrum of cholesterol (rm = 300 ms) is shown in Figure 10.19. Positive contours are shown in black and negative contours in gray. Note that the diagonal peaks are positive and extremely strong at the contour threshold used to show the weak negative crosspeaks. Streaks of 11 noise can be seen extending down from the singlet methyl peaks (H18 and H19) on the diagonal. The cholesterol structure is shown in Figure 10.20 with some of the NOE interactions indicated with double arrows. The distances in angstrom, taken from the X-ray crystal structure, are shown in Figure 10.20 next to the arrows. Along the NOESY diagonal at the lower left side are the H4eq (doublet) and H4ax (triplet) peaks around 2.2 ppm, followed (moving up and to the right side) by the H12eq and H7eq diagonal peaks (Fig. 10.19). Moving up vertically from the H4ax peak on the diagonal, we see a ZQ artifact at F1 = H7eq (long range /-coupling CH-C=C-CH) and NOE peaks at F1 = H2ax and F1 = H19 methyl. Both of these are 1,3-diaxial relationships

Figure 10.20

in the A ring of the steroid. Moving to the right on the F1 = H19 methyl line, we encounter H1eq (1,2-cis relationship) and then H11ax and H8 (1,3-diaxial). Moving up from the H12eq peak on the diagonal, we encounter H12ax (ZQ artifact), H21 methyl of the C-17 side chain, and the angular methyl group H18 (1,2-cis). Moving to the right on the F1 = H18 methyl line, we see a number of NOE interactions: H8 (1,3-diaxial), H20 of the C-17 side chain, H15^ and H16^ in the five-membered D ring, and H21 methyl of the C-17 side chain. The NOE crosspeaks from the angular methyl groups are partially obscured by t1 noise streaks below the diagonal. As ti noise is always a vertical streak (along the Fi dimension), if a crosspeak is obscured by the streak, we can always look to the other side of the diagonal to find an equivalent crosspeak that is not in the path of a t1 noise streak.

Moving to the right from the H7eq diagonal peak, we see strong ZQ artifacts at the F2 = H7ax and F2 = H8 positions. An especially strong pair of ZQ artifacts appears (center) due to the H15a-H16^ coupling. These can be seen clearly as star-shaped antiphase peaks in the expanded region shown in Figure 10.21. The center of the peak as well as four spots extending diagonally from the center are negative; four spots above, below, right, and left of the center are positive.

The 600 MHz ROESY spectrum of cholesterol is shown in Figure 10.22 (tm = 200 ms, spin-lock yB1/2n = 3333 Hz). The strong positive diagonal, weak negative crosspeaks, and t1 noise streaks coming down from the methyl diagonal peaks are all similar to the NOESY spectrum (Fig. 10.19), but the spectrum is cleaner overall with fewer ZQ artifacts. Cross-peaks can be identified from the olefinic proton H6 to H4eq (strong) and H4ax (weak) as well as to H7eq and H7ax (Fig. 10.22). From H3 we see crosspeaks to H1ax (1,3-diaxial relationship) and to the /-coupled protons H2eq, H4ax, and H4eq. The expanded upfield region of the ROESY spectrum (Fig. 10.23) can be directly compared to the NOESY (Fig. 10.19) with most of the same correlations identified corresponding to the distances shown on the structure (Fig. 10.20). Figure 10.24 shows three enlarged strips of the ROESY with F2 slices at F1 = 0.68 (H18), 1.01 (H19), and 5.35 (H6) ppm. In the slices you can clearly see the "triplet" structure of H4ax (middle slice) and the "doublet" structure of H4eq (bottom slice). Even the smaller coupling from H4eq to H3 is resolved in the bottom slice. The H4ax cross-peak dominates in the F1 = H19 slice, whereas the H4eq crosspeak is much larger in the F1 = H6 slice, similar to the results we saw in the selective 1D NOE experiment (Chapter 8, Fig. 8.36). In the F1 = H6 slice (bottom), the H7eq crosspeak clearly has doublet structure and the H7ax crosspeak has double-doublet structure. These multiplets are low resolution

Figure 10.21

Figure 10.21

and show only the large couplings—in general, only the large axial-axial and geminal (2^hh) couplings are well resolved. From the H18 methyl group (top slice), we see a strong ROE to H8 but only a weak ROE to H11ax; from the other angular methyl group H19 (middle slice), we see strong ROEs to both of these ^-axial protons. The methyl groups also "see" equatorial protons H12eq (from H18) and H1eq (from H19), which appear as doublets. From H18 we see a "quartet" structure for H16^ and a sharp doublet for the H21 methyl group. A crosspeak to H18 in the F1 = H19 slice may be due to spin diffusion through H8 or H11ax.

Figure 10.23

In cholesterol we can see nearly all of the 1,3-diaxial and 1,2 axial-equatorial NOE interactions, whether they are H-H or H-CH3 relationships. One should take great care, however, in establishing regiochemistry or stereochemistry by NOE experiments. Even in a small ring (4-6 members), the difference in the H-H distance between a cis 1,2 (vicinal) relationship and a trans relationship is small. For an ideal cyclohexane chair conformation, for example, the vicinal distances are 2.54 A for equatorial-equatorial (trans), 2.48 A for

Figure 10.25

equatorial-axial (cis), and 3.09 A for axial-axial (trans). The close 1,3 diaxial (cis) distance is characteristic for cyclohexane: 2.77 A compared to 3.89 for ax-eq and 4.35 for eq-eq. NOE interactions across the ring (1,4) are rarely observed, with distances of 4.27 (cis), 4.15 (trans diaxial), and 5.04 (trans diequatorial).

The 200-ms 600 MHz ROESY spectrum of the glycopeptide Tyr-Thr-Gly-Phe-Leu-Ser(Lactose) in 90% H2O/10% D2O is shown in Figure 10.25. Crosspeaks that are also found in the TOCSY spectrum (i.e., which are within the same amino acid residue: Fig. 9.45) are enclosed in rectangles. Negative peaks are shown in gray, and positive peaks in black. We saw in Chapter 9 how the spin systems corresponding to each amino acid residue can be identified in the 2D TOCSY spectrum from the HN resonance in F2. Now looking at the same region of the ROESY spectrum, we can see numerous NOE connections between one residue and its neighbor in the primary sequence. These are called sequential (or i ^ i + 1) NOEs because they connect across one peptide bond to the next residue in the sequence.

Starting with the F1 chemical shift of the Tyr-1 Ha proton (Fig. 10.25, right side, F1 = 4.32 ppm), we can "walk" through the peptide backbone by using the proximity of the Ha proton of residue i to the HN proton of residue i + 1: CH^ — COi — NHi+1. There is no HN chemical shift for Tyr-1 because it is the N-terminal residue and its amine group protons (not an amide) are in very fast exchange with water. Moving all the way to the left side on the F1 = Ha of Tyr-1 line, we encounter a negative crosspeak that is not found in the TOCSY spectrum: a sequential NOE to HN of Thr-2. Moving up from the NOE crosspeak leads to two weak ZQ artifacts at the same positions as the Ha and H^ shifts of Thr-2 in the TOCSY spectrum (rectangles). Moving to the right from these crosspeaks leads to two strong sequential NOE crosspeaks at the F2 chemical shift of the HN of Gly-3. In this case, both the Ha proton and the H proton of residue 2 are close to the HN of residue 3. Moving up, we come to a messy ZQ artifact at the Ha shift of Gly-3 in F1. From here, we move all the way to the right to a nice, fat sequential NOE peak at F2 = 8.01 ppm, the HN proton of Phe-4. Below this is the ZQ arifact-distorted Ha peak and above this are the Hp and Hp/ peaks of the Phe-4 spin system. From any of these three "intraresidue" crosspeaks, one can move to the left and run into a sequential NOE peak at the HN chemical shift of Leu-5 in F2. Again, not just the Ha proton of Phe-4 but also the Hp and Hp protons are all close enough in space to the HN of Leu-5 to give ROESY crosspeaks. Looking along this vertical line (F2 = Hn of Leu-5), we find the ZQ artifact corresponding to F1 = Ha of Leu-5 (rectangle) and moving to the right a very short distance there is a strong sequential NOE crosspeak at the chemical shift of HN of Ser-6 in F2. Above this crosspeak are the intraresidue F1 = Hp and F1 = Hp of Ser-6 crosspeaks, and below this is the F1 = Ha intraresidue crosspeak of Ser-6. This completes our "walk" along the backbone. In this case, the assignments can be made from the TOCSY alone because each spin system has a unique pattern of chemical shifts that makes it easy to identify in the known primary sequence. But in more complex peptides and proteins, this "walk" is a way to make sequence-specific assignments even when there may be several examples of each of the 20 amino acids in the sequence.

10.3.6 Distance Measurement From NOESY

Ideally, the initial rate of increase of the NOE intensity as a function of mixing time is directly proportional to 1 /r^D, where rab is the distance between Ha and Hb. There are a number of caveats for those who wish to measure distances using crosspeak intensities in a NOESY spectrum. First, the crosspeak intensities (volumes) are in arbitrary units so that we must calibrate them with a pair (or better yet several pairs) of protons with an accurately known distance within the molecule. This can be done with a geminal pair, a vicinal pair on an aromatic ring, or a 1,3-diaxial pair in a clearly-defined cyclohexane chair structure. Second, the above discussion assumed that there are only two nuclei, Ha and Hb, related by an NOE interaction. This is called the isolated spin-pair hypothesis. In reality, it is very rare to find two protons that have no other neighbors within 5 A. Usually this distance includes other protons, which are related to still others by NOE interactions. The result is a process called spin diffusion, where perturbation of the populations (z magnetization) of one nucleus affects the populations of nearby nuclei, which in turn perturb the populations of their neighbors in an expanding process. Spin diffusion can lead to crosspeaks between pairs of nuclei that are not directly related by an NOE interaction and may be considerably more than 5 A apart in space. Because for small molecules the NOE transfer leads to a reversal of sign of the z magnetization, spin diffusion involving two steps will give crosspeaks that are positive with respect to the diagonal, so that these can be spotted easily. The best way to avoid spin diffusion, however, is to choose as short a mixing time (rm) as possible. This also increases the accuracy of distance measurements because only for short mixing times is the crosspeak intensity truly proportional to the cross-relaxation rate and thus to 1/r6. Sometimes a series of NOESY spectra is acquired using a number of different mixing times, and the crosspeak intensities are plotted as function of tm. This NOESY buildup study allows the initial slope of the NOE buildup curve to be measured directly as the cross-relaxation rate. Spin-diffusion crosspeaks can be identified in a buildup study because they will always show a lag (sigmoidal shape) before crosspeak intensity starts to climb.

All of these considerations would indicate that a short mixing time is good. Unfortunately, short mixing times also mean weak crosspeaks that are near the noise level. Larger numbers of transients will be required for short mixing times, which means longer overall experiment times. If you are not doing quantitative distance measurements, the mixing time is usually set somewhere near the T1 value (characteristic time for simple self-relaxation). For these routine experiments, we are shooting for a maximum NOE intensity so that intensity is not strictly proportional to 1/r6 and spin diffusion has to be considered as a possible complicating factor in the interpretation of crosspeaks. In this case, we should classify NOE crosspeaks as "weak," "medium," or "strong" and not make any attempt to measure accurate distances.

Because of all of the above pitfalls, NOE is probably the most misinterpreted experiment in organic chemistry. In my experience, /-coupling measurements, both homonuclear and heteronuclear, give far more reliable information than NOE measurements in the determination of small-molecule stereochemistry. To use NOE measurements for stereochemical determinations, it is always best to do the NOESY experiment on both isomers and compare the crosspeak intensities (relative to the diagonal peak intensities) and measure distances on both isomers using an energy-minimized computer model of the structures. If the differences in distance and NOE intensity are small between the two isomers, the experiment cannot be conclusive.

10.3.7 Baseline Correction

Just as a rolling baseline will affect the integral area of a peak in a 1D spectrum, baseline errors (or "baseplane errors") in a 2D spectrum will render crosspeak volumes inaccurate. For any quantitative distance determinations, the baseline must be very flat with no streaks or artifacts passing through the crosspeak. One way to check for baseline errors is to measure the volume of two crosspeaks that are symmetry-related across the diagonal—they should be of similar intensity. Another way is to measure the volume within a rectangle in the noise (with no crosspeaks) and compare it to a similar rectangle ("footprint") on a crosspeak. The noise volume should be very small compared to the crosspeak volume. Another way is to display a 1D slice (row or column) from the 2D data matrix and display a zero data set on top of it in a contrasting color. When the vertical scale is increased, you will see where the baseline (noise level where there are no peaks) deviates from the zero line.

10.3.8 EXSY: Chemical Exchange in 2D NMR

Crosspeaks in a NOESY spectrum can also arise from exchange. Even if exchange is slow on the NMR timescale (tc = V2/(nAv)), we can observe it in a longer timescale, the mixing time of the NOESY experiment. If you see positive crosspeaks (relative to a positive diagonal) in the NOESY spectrum of a small molecule (as always, "small" means anc ^ 1), these crosspeaks represent exchange rather than an NOE interaction. Exchange means that a nucleus in one environment physically moves to another environment with a different chemical shift. If the rate is much less than the coalescence time tc = 1/(2.22 x Av), two sharp lines will still be observed in the 1D spectrum at the two chemical-shift positions ("slow exchange"). If, for example, half of the Ha nuclei in a 2D NOESY experiment undergo exchange to the Hb position during the mixing time tm, we have

-Ii!cos(^ai1)(^) -0.5 Ii!cos(^a¿1) -0.5 Ibcos(^a¿0

Diagonal Crosspeak where I^ and Ib represent z magnetization in the two environments. Effectively, the Ha nucleus has changed into an Hb nucleus (changed its chemical shift from to and because the chemistry did not affect the nuclear spin, it will have exactly the same nuclear spin state when it becomes Hb magnetization. When a Ha nucleus "turns into" an Hb nucleus, it "drags" its spin state over from chemical shift Qa to chemical shift Qb. In the slow exchange case, we may still be able to observe the exchange in a NOESY spectrum because the proton changes its environment during the much longer timescale of the mixing time (typically hundreds of milliseconds versus a few milliseconds for tc). This term will lead to a crosspeak at F1 = Qa, F2 = which has the same sign as the diagonal. For small molecules, exchange peaks stand out clearly because they are opposite in sign to the NOE peaks. Because both NOE and exchange crosspeaks show up in this experiment, NOESY was named nuclear Overhauser and exchange spectroscopy. Sometimes if a NOESY experiment is run solely for the purpose of studying exchange, it is called "EXSY" in the literature (Exchange Spectroscopy), but the experiment is identical to a NOESY experiment. For large molecules, exchange will not stand out in the NOESY spectrum as NOE crosspeaks are also positive. In this case, a ROESY spectrum will allow us to differentiate exchange from NOE because the NOE crosspeaks are all negative in a ROESY spectrum and the exchange crosspeaks are positive.

Figure 10.26 shows the 300-ms NOESY spectrum of lactose (Fig. 10.9) in D2O/NaOD. Negative contours are shown in gray and positive contours in black; the diagonal is phased

Figure 10.26

Figure 10.26

to positive intensity. In the lower left corner, we see the diagonal peak for the H-1 proton of the glucose portion of a-lactose (a-glu-1). Moving up on the vertical line, we encounter at F1 = j-glu-1 (4.54 ppm) a positive, in-phase crosspeak. This is the exchange peak for conversion of the j-glu-1 proton to an a-glu-1 proton during the mixing time. The inset above the crosspeak shows the 1D horizontal slice through this cross-peak. Moving to the right along the Fi = j-glu-1 horizontal line, we pass the diagonal peak and move to the right to a pair of negative (NOE) crosspeaks around F2 = 3.5 ppm. These are NOE peaks from j-glu-1 (axial proton) to H-3 and H-5 of the glucose portion of j-lactose (Fig. 10.9), representing 1,3-diaxial relationships. The inset below the crosspeaks shows a horizontal slice: the peaks are clearly in-phase and negative. Moving farther to the right on the F1 = j-glu-1 line we come to the j-glu-2 crosspeak (F2 = 3.15 ppm), which is distorted by the ZQ (/-coupling) artifact. Similar NOE crosspeaks are seen starting from gal-1 on the diagonal, but no exchange peak is observed because the galactose anomeric position is locked in the ^-orientation by the glycosidic linkage to the glucose-4 position. On the F1 = a-glu-1 horizontal line, we see the positive exchange peak with j-glu-1 and the (ZQ artifact-distorted) NOE crosspeak with a-glu-2. Because the a-glu-1 proton is equatorial, we do not see any NOE crosspeaks to a-glu-3 or a-glu-5.

10.4 EXPANDING OUR VIEW OF COHERENCE: QUANTUM MECHANICS AND SPHERICAL OPERATORS

10.4.1 Double-Quantum and Zero-Quantum Coherence

Single-quantum coherence (SQC) can be easily defined in terms of the vector model. In a population of identical spins, each individual spin precesses in the laboratory magnetic field at the same frequency, the Larmor frequency vo. At equilibrium, the orientation of the spins on the "cone" of precession is random: they are spread out evenly around the cone at any instant in time. Thus the x and y components of their magnetization cancel perfectly leaving no net magnetization in the x-y plane: Mx = 0 and My = 0. After a 90° pulse, the spins become "organized" such that they are now rotating "in phase"—at any instant in time they are all at the same orientation on the cone. Because their phases are coherent instead of random, we call this phase coherence or simply coherence. Now the individual x and y components of magnetization add together instead of canceling, resulting in a net magnetization in the x-y plane that rotates at the Larmor frequency. It is this net magnetization (coherence) that induces a sinusoidal voltage in the probe coil, which is recorded by the spectrometer as the FID.

There are, however, other kinds of coherence that play an important role in many NMR experiments, such as DEPT, DQF-COSY, HMQC, and HMBC. Coherences other than SQC are called multiple-quantum coherences (MQC), including zero-quantum and doublequantum coherences (ZQC and DQC). When we are talking about SQC, we are referring to NMR transitions that involve only one spin, changing its orientation with respect to the Bo field. For example, in a two-spin system (1H-13C) we can talk about a transition from aHaC to jHaC in which the proton changes from the a to the j state whereas the carbon remains in the a state. This transition corresponds to SQC, as only the proton undergoes a change in orientation. It is an observable transition, giving rise to one of the two components of the proton doublet in the proton spectrum.

Suppose that we are talking about a double-quantum transition in which both the proton and carbon change from the a state to the j state. This transition is thus from the aHaC state to the jH jC state of the two-spin, four-state system. This transition corresponds to DQC. Likewise, if the proton flips from j to a while the carbon simultaneously flips from a to j, we have a zero-quantum transition (jHaC to aH jC) because the total number of spins in the excited (j) state has not changed. This transition corresponds to ZQC. What can we say about these mysterious coherences? In Section 7.10, we encountered ZQC and DQC as intermediate states in coherence transfer, created with pulses from antiphase SQC:

Note that in the product operator —2IxSy we have both spins, I and S (1H and 13C) in the x-y plane, with the operators multiplied together. This means that both spins are undergoing transitions at the same time, so we have ZQC and DQC. We can convert ZQC and DQC back into observable SQC with a second pulse

(The factor of 4 reflects the change from observing1H to observing 13 C, a change in our standard of comparison for magnitude). The net effect of these two steps is to convert antiphase proton SQC into antiphase carbon SQC, an overall coherence transfer with ZQC/DQC as an intermediate state. In Section 7.11 the product operator representations of pure ZQC and DQC were introduced, and we saw that pure DQC rotates in the x-y plane just like SQC, but at a frequency that is the sum of the frequencies of the two spins involved:

{DQ}x ^{DQjy ^ -{DQ}x ^ -{DQ}y ^{DQjx frequency of precession = vH - vC

For example, on a 600-MHz spectrometer XH SQC precesses at 600 MHz, 13C SQC pre-cesses at 150 MHz, and {XH-13C} DQC precesses at 750 MHz. This makes sense because we are talking about a transition in which both XH and 13 C change from the a to the j state. Zero quantum coherence behaves in a similar way, but the precession rate is the difference between the two SQ precession frequencies:

{ZQ}x ^{ZQ}y ^ -{ZQ}x ^ -{ZQ}y ^{ZQ}x frequency of precession = vH - vC

Thus on the same 600-MHz spectrometer, {1H-13C} ZQC precesses at 450 MHz (600-150). The good news is that there is no /-coupling evolution due to the active 1H-13C J coupling. Because both spins are undergoing transitions, the interaction between the magnetic dipoles of the two spins does not change: If they are aligned (aa or jj), they remain aligned in DQC; if they are opposite (aj or ja), they remain opposite in ZQC. Passive couplings (coupling to nuclei other than these two, e.g., I' or S') will lead to /-coupling evolution and multiplication by Iz' or Sz operators.

The pairwise products of the operators Ix, Iy, Sx, and Sy can be expressed in terms of pure ZQC and DQC as follows:

2IXSX = 0.5({DQ}x + {ZQ}x) 2IySy = 0.5({ZQ}x - (DQ}X)

2IxSy = 0.5({DQ}y + {ZQ}y) 2Iy Sx = 0.5({DQ}y - {ZQ}y)

Now you can see why the conversion of antiphase XH SQC with a 90° pulse results in a mixture of DQC and ZQC:

90Xon13C

These multiple quantum coherences cannot be visualized with the vector system, even though we can talk about them as being on the x axis or the y axis. So what do they represent and how can we think about them?

10.4.2 Coherence: The Quantum View

To understand any coherence other than SQC, we need a new and more general definition of coherence. Coherence arises from the quantum mechanical mixing or overlap of spin states ("superposition"). In the two spin system (I, S = XH, 13C) we have four spin states (aa, aj, ja, and ¡30), which are all stable states of defined energy. Let's talk about a single 1H-13C pair (one molecule). It is possible for this pair to be in any one of the four energy states, but it is also possible for the pair to be in a mixture or overlap or superposition of two states. This is one of the fundamental tenets of quantum mechanics: Sometimes you cannot be sure which energy state a particle is in. Let's say that this particular pair is in a mixture of states aa and jj :

The spin state (or "wave function") of this pair is a linear combination of the states aa and 33, with coefficients c1 and c2. These coefficients are actually complex numbers, with real parts (a) and imaginary parts (b):

c1 = a1 + b1i c4 = a4 + b4i where i is the square root of -1 (i2 = -1). We cannot say which energy state the pair is in, but we can talk about probabilities. The probability of the pair being in the aa state is

P(aa) = c*c1 = (a1 - b1i)(a1 + b1i) = a2 - b2i2 = a2 + b?

where c* is the complex conjugate of c 1. Note that P(aa) is a positive real number, and if the coefficients are properly normalized, it will be between 0 and 1. Likewise, the probability of this spin pair being in the 33 state is

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