Using 12V DC to create symmetric power source

You would do much better by using an opamp to make the ground point.

I sugget using a circuit like this:

Just use your 12 V instead of the 9V battery. It depends on the opamp how much current can flow through the ground connection.

This circuit is also much more efficient than your proposal.

To make it suitable for higher (ground) currents, use a power-opamp (some audio power amps are also suitable) or see WhatRoughBeats's answer for a discrete solution. Note that that solution has no over current / over temperature protection.

Well, if you reverse the zeners it will work in principle. In practice, not so well. The problem is that zeners come with a tolerance, with about 5% as the norm. So your nominal 5.6 zeners, which drop a total of 11.2 volts, could produce a "real" voltage of anywhere from 11 to 12.2 volts. The larger voltage, obviously, will not provide good regulation, and the lower voltage will draw more current from the supply and dissipate more power in the resistor.

Rather worse is the effect of that 100 ohm dropping resistor. Since all your op amp current flows through it, any current at all will affect the supply voltages, and this is a recipe for your op amps oscillating. A total of 10 mA, for instance, will drop 1 volt in the resistor and the zeners will be completely ineffective.

A better virtual ground circuit would be something like

^{simulate this circuit – Schematic created using CircuitLab} For the transistors shown, a virtual ground current of about +/- 50 mA seems reasonable, which is about 300 mW in the affected transistor.

As opposed to using a regulator (which is also an option), this will keep the virtual ground centered between the power supply + and -, which you may find preferable.

^{simulate this circuit – Schematic created using CircuitLab}

Figure 1. Regular 9 V PSU wall-wart. Figure 2. Modified for +9/0/-9 V.

You can easily modify a standard wall wart supply as shown in Figure 2. Ripple voltage will be worse and max current on each supply will be half of original specification so I'd recommend some large caps or voltage regulators to eliminate hum.

If you want to keep a standard jack on the PSU then convert it to an AC PSU by removing the rectifier and capacitors and put the diodes and caps into your project case.