Light aircraft Instructor Pilot reported receiving a low altitude alert from ATC in the vicinity of TRI airport. Temporary loss of communication with ATC was a factor.

Narrative

Aircraft was operating on an IFR flight plan in visual meteorological conditions to conduct a series of practice instrument approaches. The aircraft was supplied by the pilot flying who was the PIC and sole manipulator of the controls during the flight; and the sole operator of the radio communications with ATC. My role was to provide evaluation and instruction as necessary for the purpose of the flying pilot to obtain the certification of Instructor Pilot. The flight was planned as a round robin flight originating at ZZZ and ending at ZZZ. Intermediate approaches were planned at GCY; TRI and MOR. After completing a practice approach at GCY (a non towered airport) the flying pilot had reestablished radio contact with TRI Cities approach as instructed during the execution of the missed approach procedure. The pilot flying requested a practice ILS approach for Runway 23 at TRI. TRI Cities advised the flying pilot to expect that approach issued vector instructions for ILS 23 at TRI. Radio traffic was extremely light and several minutes passed between communications between the controller and any other aircraft on frequency. While following a vector on the right downwind the pilot flying was instructed to contact approach on a different frequency. The pilot flying changed the radio frequency in use and reported on the new frequency. The flying pilot was busy checking the ATIS as well as setting the TRI Tower frequency into standby on Com 1. Additionally the flying pilot was executing a partial panel approach with a simulated PFD (primary flight display) failure. As the aircraft reached a point a few miles from the anticipated right base vector I became concerned we hadn't heard a subsequent radio call and queried the flying pilot to verify he hadn't switched us off the correct frequency. After reviewing the approach chart he discovered he had made an entry error in the avionics. The pilot flying switched to the correct frequency and immediately advised ATC he had inadvertently been on the wrong frequency. ATC advised him he was below the MVA and instructed him to climb 600' and turn left to avoid terrain. The pilot flying immediately acknowledged and complied with ATC's instructions. We were operating in VMC and had more than 1000' of clearance to any terrain or obstacle within several miles of the aircraft. At no time was the aircraft in danger of imminent collision with any terrain or obstacles. ATC advised the pilot flying of possible pilot deviation due to operation of the aircraft below MVA while on an IFR fight plan. Post flight debrief discussion and review revealed the pilot read back the correct frequency; but incorrectly entered the frequency in the avionics. Neither the pilot flying or myself realized ATC did not acknowledge the pilot's frequency change. Intercom communication as well as the pilots approach and landing workload interfered with early identification of the data error. A contributing factor was the unusually quiet radio communications with any other aircraft during the early vectors. This resulted in the pilots not immediately noticing a state of change with the current communications with ATC. In the future no frequency change will be assumed to have been correctly made unless positive acknowledgment by ATC has been received. I was not making the radio calls; and my expectation bias led me to believe the pilot flying would have made a subsequent verification call if he had not received acknowledgment of a correct frequency switch.